[{"data":1,"prerenderedAt":21340},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-05-30-about-the-tensor":3,"i-ci:tag":21334,"i-mingcute:time-line":21338},{"post":4,"nextPost":5978,"prevPost":21009},{"id":5,"title":6,"body":7,"description":5952,"draft":5964,"enableComment":5965,"extension":5966,"image":5952,"important":5964,"location":5967,"meta":5968,"navigation":5965,"ogImage":5967,"path":5969,"seo":5970,"stem":5971,"summary":5972,"tags":5973,"time":5976,"__hash__":5977},"blog\u002Fblog\u002F2026\u002F2026-05-30-about-the-tensor.md","张量 Tensor 简述",{"type":8,"value":9,"toc":5951},"minimark",[10,17,20,23,26,29,33,36,48,55,107,110,113,116,402,405,761,764,1069,1072,1112,1153,1311,1314,1317,1320,1323,1473,1476,1961,1964,1972,1975,2060,2063,2066,2069,2168,2171,2174,2287,2295,2447,2513,2516,2577,2684,2794,2902,2905,2908,2999,3002,3184,3187,3354,3409,3412,3415,3565,5212,5215,5523,5526,5529,5532,5918,5921,5924,5936,5939,5942,5945,5948],[11,12,13],"blockquote",{},[14,15,16],"p",{},"张量是标量、向量和矩阵的高维推广",[14,18,19],{},"你几乎肯定见过这句话。它出现在几乎所有教科书的第一页，也几乎是所有困惑的出发点。",[14,21,22],{},"原因在于，这句话只描述了张量的存储格式，而对它的本质只字未提。这就好比说「人是一堆细胞的集合」——不算错，却没有回答「人到底是什么」这个真正的问题。",[14,24,25],{},"本文只追求一个目标：为什么物理学家和数学家如此坚持区分向量与「对偶向量」，并坚持区分张量中的「上标」与「下标」？只要你能抓住这一点，围绕张量的所有神秘感都会消散。",[14,27,28],{},"我们按以下顺序展开：先重新审视向量本身；接着介绍它的孪生兄弟——对偶向量；然后给出张量的严格定义；用一个具体例子拆解一个常见的困惑点；最后阐明连接向量与对偶向量的那座关键桥梁——度量。",[30,31,32],"h2",{"id":32},"箭头",[14,34,35],{},"假设平面上有一支物理意义上的箭头，指向北偏东 30°，长 5 米。这支箭头具有客观存在性——无论你如何建立坐标系，它都在原处；它的长度和方向并不会因为你的参考系变了就改变。",[14,37,38,39,43,44,47],{},"然而，当你写下 ",[40,41,42],"code",{},"(3, 4)"," 这样的坐标时，你已经做了一次隐蔽的操作：你选择了一个坐标基，然后把这个箭头相对于该基「翻译」成了一组数字。换一个基（比如把坐标轴旋转 45°），同一支箭头会被翻译成完全不同的数字，比如 ",[40,45,46],{},"(4.95, 0.71)","。",[14,49,50,51,54],{},"数字变了，箭头没变。这正是向量作为几何对象与其坐标表示（数组）之间最根本的区别。深度学习里的张量（例如 ",[40,52,53],{},"torch.tensor([3,4])","）只是这个「翻译结果」，从不关心背后是否站着一个不变的几何对象。纯粹做数值计算时这完全没问题，但一旦你想谈论「物理定律不依赖于观察者的选择」这类命题，就必须回到箭头本身。",[14,56,57,58,106],{},"抛开坐标不谈，向量空间 ",[59,60,64,86],"span",{"className":61,"translate":63},[62],"katex","no",[59,65,68],{"className":66},[67],"katex-mathml",[69,70,72],"math",{"xmlns":71},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[73,74,75,82],"semantics",{},[76,77,78],"mrow",{},[79,80,81],"mi",{},"V",[83,84,81],"annotation",{"encoding":85},"application\u002Fx-tex",[59,87,91],{"className":88,"ariaHidden":90},[89],"katex-html","true",[59,92,95,100],{"className":93},[94],"base",[59,96],{"className":97,"style":99},[98],"strut","height:0.6833em;",[59,101,81],{"className":102,"style":105},[103,104],"mord","mathnormal","margin-right:0.2222em;"," 的定义其实相当简洁：只要一个对象集合支持加法（首尾相接的合成）和标量乘法（拉伸或压缩），并满足少数几条自然的代数定律，它就构成一个向量空间，其元素就是向量。速度、力、位移都是这类对象。",[30,108,109],{"id":109},"对偶空间",[14,111,112],{},"向量是箭头；那么，「测量」这支箭头的工具又是什么？",[14,114,115],{},"它是一种完全不同类型的对象，叫作对偶向量（也叫余向量或 1-形式）。它不是箭头，而是一把标尺——一组等距的平行线。把箭头放上去，箭头穿过的线数就是它输出的数字。",[14,117,118,119,194,195,223,224,282,283,315,316,350,351,401],{},"严格定义：对偶空间 ",[59,120,122,143],{"className":121,"translate":63},[62],[59,123,125],{"className":124},[67],[69,126,127],{"xmlns":71},[73,128,129,140],{},[76,130,131],{},[132,133,134,136],"msup",{},[79,135,81],{},[137,138,139],"mo",{},"∗",[83,141,142],{"encoding":85},"V^*",[59,144,146],{"className":145,"ariaHidden":90},[89],[59,147,149,153],{"className":148},[94],[59,150],{"className":151,"style":152},[98],"height:0.6887em;",[59,154,156,159],{"className":155},[103],[59,157,81],{"className":158,"style":105},[103,104],[59,160,163],{"className":161},[162],"msupsub",[59,164,167],{"className":165},[166],"vlist-t",[59,168,171],{"className":169},[170],"vlist-r",[59,172,175],{"className":173,"style":152},[174],"vlist",[59,176,178,183],{"style":177},"top:-3.063em;margin-right:0.05em;",[59,179],{"className":180,"style":182},[181],"pstrut","height:2.7em;",[59,184,190],{"className":185},[186,187,188,189],"sizing","reset-size6","size3","mtight",[59,191,139],{"className":192},[193,189],"mbin"," 是 ",[59,196,198,211],{"className":197,"translate":63},[62],[59,199,201],{"className":200},[67],[69,202,203],{"xmlns":71},[73,204,205,209],{},[76,206,207],{},[79,208,81],{},[83,210,81],{"encoding":85},[59,212,214],{"className":213,"ariaHidden":90},[89],[59,215,217,220],{"className":216},[94],[59,218],{"className":219,"style":99},[98],[59,221,81],{"className":222,"style":105},[103,104]," 上所有线性泛函的集合。也就是说，",[59,225,227,244],{"className":226,"translate":63},[62],[59,228,230],{"className":229},[67],[69,231,232],{"xmlns":71},[73,233,234,242],{},[76,235,236],{},[132,237,238,240],{},[79,239,81],{},[137,241,139],{},[83,243,142],{"encoding":85},[59,245,247],{"className":246,"ariaHidden":90},[89],[59,248,250,253],{"className":249},[94],[59,251],{"className":252,"style":152},[98],[59,254,256,259],{"className":255},[103],[59,257,81],{"className":258,"style":105},[103,104],[59,260,262],{"className":261},[162],[59,263,265],{"className":264},[166],[59,266,268],{"className":267},[170],[59,269,271],{"className":270,"style":152},[174],[59,272,273,276],{"style":177},[59,274],{"className":275,"style":182},[181],[59,277,279],{"className":278},[186,187,188,189],[59,280,139],{"className":281},[193,189]," 中的每个元素 ",[59,284,286,301],{"className":285,"translate":63},[62],[59,287,289],{"className":288},[67],[69,290,291],{"xmlns":71},[73,292,293,298],{},[76,294,295],{},[79,296,297],{},"ω",[83,299,300],{"encoding":85},"\\omega",[59,302,304],{"className":303,"ariaHidden":90},[89],[59,305,307,311],{"className":306},[94],[59,308],{"className":309,"style":310},[98],"height:0.4306em;",[59,312,297],{"className":313,"style":314},[103,104],"margin-right:0.0359em;"," 都是一条规则：给它一个向量 ",[59,317,319,335],{"className":318,"translate":63},[62],[59,320,322],{"className":321},[67],[69,323,324],{"xmlns":71},[73,325,326,332],{},[76,327,328],{},[79,329,331],{"mathvariant":330},"bold","v",[83,333,334],{"encoding":85},"\\mathbf{v}",[59,336,338],{"className":337,"ariaHidden":90},[89],[59,339,341,345],{"className":340},[94],[59,342],{"className":343,"style":344},[98],"height:0.4444em;",[59,346,331],{"className":347,"style":349},[103,348],"mathbf","margin-right:0.016em;","，它输出一个实数 ",[59,352,354,377],{"className":353,"translate":63},[62],[59,355,357],{"className":356},[67],[69,358,359],{"xmlns":71},[73,360,361,374],{},[76,362,363,365,369,371],{},[79,364,297],{},[137,366,368],{"stretchy":367},"false","(",[79,370,331],{"mathvariant":330},[137,372,373],{"stretchy":367},")",[83,375,376],{"encoding":85},"\\omega(\\mathbf{v})",[59,378,380],{"className":379,"ariaHidden":90},[89],[59,381,383,387,390,394,397],{"className":382},[94],[59,384],{"className":385,"style":386},[98],"height:1em;vertical-align:-0.25em;",[59,388,297],{"className":389,"style":314},[103,104],[59,391,368],{"className":392},[393],"mopen",[59,395,331],{"className":396,"style":349},[103,348],[59,398,373],{"className":399},[400],"mclose","，并且这条规则对向量加法和标量乘法都是线性的。",[14,403,404],{},"在有限维线性代数中，这简单到几乎平庸：",[406,407,408,580],"ul",{},[409,410,411,412,440,441],"li",{},"向量 ",[59,413,415,428],{"className":414,"translate":63},[62],[59,416,418],{"className":417},[67],[69,419,420],{"xmlns":71},[73,421,422,426],{},[76,423,424],{},[79,425,331],{"mathvariant":330},[83,427,334],{"encoding":85},[59,429,431],{"className":430,"ariaHidden":90},[89],[59,432,434,437],{"className":433},[94],[59,435],{"className":436,"style":344},[98],[59,438,331],{"className":439,"style":349},[103,348]," 是列向量：",[59,442,444,491],{"className":443,"translate":63},[62],[59,445,447],{"className":446},[67],[69,448,449],{"xmlns":71},[73,450,451,488],{},[76,452,453,456,485],{},[137,454,455],{"fence":90},"[",[457,458,462,476],"mtable",{"rowspacing":459,"columnalign":460,"columnspacing":461},"0.16em","center","1em",[463,464,465],"mtr",{},[466,467,468],"mtd",{},[469,470,472],"mstyle",{"scriptlevel":471,"displaystyle":367},"0",[473,474,475],"mn",{},"2",[463,477,478],{},[466,479,480],{},[469,481,482],{"scriptlevel":471,"displaystyle":367},[473,483,484],{},"3",[137,486,487],{"fence":90},"]",[83,489,490],{"encoding":85},"\\begin{bmatrix} 2 \\\\ 3 \\end{bmatrix}",[59,492,494],{"className":493,"ariaHidden":90},[89],[59,495,497,501],{"className":496},[94],[59,498],{"className":499,"style":500},[98],"height:2.4em;vertical-align:-0.95em;",[59,502,505,514,574],{"className":503},[504],"minner",[59,506,510],{"className":507,"style":509},[393,508],"delimcenter","top:0em;",[59,511,455],{"className":512},[513,188],"delimsizing",[59,515,517],{"className":516},[103],[59,518,520],{"className":519},[457],[59,521,524],{"className":522},[523],"col-align-c",[59,525,528,565],{"className":526},[166,527],"vlist-t2",[59,529,531,560],{"className":530},[170],[59,532,535,548],{"className":533,"style":534},[174],"height:1.45em;",[59,536,538,542],{"style":537},"top:-3.61em;",[59,539],{"className":540,"style":541},[181],"height:3em;",[59,543,545],{"className":544},[103],[59,546,475],{"className":547},[103],[59,549,551,554],{"style":550},"top:-2.41em;",[59,552],{"className":553,"style":541},[181],[59,555,557],{"className":556},[103],[59,558,484],{"className":559},[103],[59,561,564],{"className":562},[563],"vlist-s","​",[59,566,568],{"className":567},[170],[59,569,572],{"className":570,"style":571},[174],"height:0.95em;",[59,573],{},[59,575,577],{"className":576,"style":509},[400,508],[59,578,487],{"className":579},[513,188],[409,581,582,583,611,612],{},"对偶向量 ",[59,584,586,599],{"className":585,"translate":63},[62],[59,587,589],{"className":588},[67],[69,590,591],{"xmlns":71},[73,592,593,597],{},[76,594,595],{},[79,596,297],{},[83,598,300],{"encoding":85},[59,600,602],{"className":601,"ariaHidden":90},[89],[59,603,605,608],{"className":604},[94],[59,606],{"className":607,"style":310},[98],[59,609,297],{"className":610,"style":314},[103,104]," 是行向量：",[59,613,615,650],{"className":614,"translate":63},[62],[59,616,618],{"className":617},[67],[69,619,620],{"xmlns":71},[73,621,622,647],{},[76,623,624,626,645],{},[137,625,455],{"fence":90},[457,627,629],{"rowspacing":459,"columnalign":628,"columnspacing":461},"center center",[463,630,631,638],{},[466,632,633],{},[469,634,635],{"scriptlevel":471,"displaystyle":367},[473,636,637],{},"1",[466,639,640],{},[469,641,642],{"scriptlevel":471,"displaystyle":367},[473,643,644],{},"4",[137,646,487],{"fence":90},[83,648,649],{"encoding":85},"\\begin{bmatrix} 1 & 4 \\end{bmatrix}",[59,651,653],{"className":652,"ariaHidden":90},[89],[59,654,656,660],{"className":655},[94],[59,657],{"className":658,"style":659},[98],"height:1.2em;vertical-align:-0.35em;",[59,661,663,670,755],{"className":662},[504],[59,664,666],{"className":665,"style":509},[393,508],[59,667,455],{"className":668},[513,669],"size1",[59,671,673],{"className":672},[103],[59,674,676,713,718,721],{"className":675},[457],[59,677,679],{"className":678},[523],[59,680,682,704],{"className":681},[166,527],[59,683,685,701],{"className":684},[170],[59,686,689],{"className":687,"style":688},[174],"height:0.85em;",[59,690,692,695],{"style":691},"top:-3.01em;",[59,693],{"className":694,"style":541},[181],[59,696,698],{"className":697},[103],[59,699,637],{"className":700},[103],[59,702,564],{"className":703},[563],[59,705,707],{"className":706},[170],[59,708,711],{"className":709,"style":710},[174],"height:0.35em;",[59,712],{},[59,714],{"className":715,"style":717},[716],"arraycolsep","width:0.5em;",[59,719],{"className":720,"style":717},[716],[59,722,724],{"className":723},[523],[59,725,727,747],{"className":726},[166,527],[59,728,730,744],{"className":729},[170],[59,731,733],{"className":732,"style":688},[174],[59,734,735,738],{"style":691},[59,736],{"className":737,"style":541},[181],[59,739,741],{"className":740},[103],[59,742,644],{"className":743},[103],[59,745,564],{"className":746},[563],[59,748,750],{"className":749},[170],[59,751,753],{"className":752,"style":710},[174],[59,754],{},[59,756,758],{"className":757,"style":509},[400,508],[59,759,487],{"className":760},[513,669],[14,762,763],{},"它们的「配对」就是矩阵乘法：",[59,765,768],{"className":766,"translate":63},[767],"katex-display",[59,769,771,846],{"className":770,"translate":63},[62],[59,772,774],{"className":773},[67],[69,775,777],{"xmlns":71,"display":776},"block",[73,778,779,843],{},[76,780,781,783,785,787,789,792,814,838,840],{},[79,782,297],{},[137,784,368],{"stretchy":367},[79,786,331],{"mathvariant":330},[137,788,373],{"stretchy":367},[137,790,791],{},"=",[76,793,794,796,812],{},[137,795,455],{"fence":90},[457,797,798],{"rowspacing":459,"columnalign":628,"columnspacing":461},[463,799,800,806],{},[466,801,802],{},[469,803,804],{"scriptlevel":471,"displaystyle":367},[473,805,637],{},[466,807,808],{},[469,809,810],{"scriptlevel":471,"displaystyle":367},[473,811,644],{},[137,813,487],{"fence":90},[76,815,816,818,836],{},[137,817,455],{"fence":90},[457,819,820,828],{"rowspacing":459,"columnalign":460,"columnspacing":461},[463,821,822],{},[466,823,824],{},[469,825,826],{"scriptlevel":471,"displaystyle":367},[473,827,475],{},[463,829,830],{},[466,831,832],{},[469,833,834],{"scriptlevel":471,"displaystyle":367},[473,835,484],{},[137,837,487],{"fence":90},[137,839,791],{},[473,841,842],{},"14",[83,844,845],{"encoding":85},"\\omega(\\mathbf{v}) = \\begin{bmatrix} 1 & 4 \\end{bmatrix} \\begin{bmatrix} 2 \\\\ 3 \\end{bmatrix} = 14",[59,847,849,879,1059],{"className":848,"ariaHidden":90},[89],[59,850,852,855,858,861,864,867,872,876],{"className":851},[94],[59,853],{"className":854,"style":386},[98],[59,856,297],{"className":857,"style":314},[103,104],[59,859,368],{"className":860},[393],[59,862,331],{"className":863,"style":349},[103,348],[59,865,373],{"className":866},[400],[59,868],{"className":869,"style":871},[870],"mspace","margin-right:0.2778em;",[59,873,791],{"className":874},[875],"mrel",[59,877],{"className":878,"style":871},[870],[59,880,882,885,980,984,1050,1053,1056],{"className":881},[94],[59,883],{"className":884,"style":500},[98],[59,886,888,894,974],{"className":887},[504],[59,889,891],{"className":890,"style":509},[393,508],[59,892,455],{"className":893},[513,669],[59,895,897],{"className":896},[103],[59,898,900,934,937,940],{"className":899},[457],[59,901,903],{"className":902},[523],[59,904,906,926],{"className":905},[166,527],[59,907,909,923],{"className":908},[170],[59,910,912],{"className":911,"style":688},[174],[59,913,914,917],{"style":691},[59,915],{"className":916,"style":541},[181],[59,918,920],{"className":919},[103],[59,921,637],{"className":922},[103],[59,924,564],{"className":925},[563],[59,927,929],{"className":928},[170],[59,930,932],{"className":931,"style":710},[174],[59,933],{},[59,935],{"className":936,"style":717},[716],[59,938],{"className":939,"style":717},[716],[59,941,943],{"className":942},[523],[59,944,946,966],{"className":945},[166,527],[59,947,949,963],{"className":948},[170],[59,950,952],{"className":951,"style":688},[174],[59,953,954,957],{"style":691},[59,955],{"className":956,"style":541},[181],[59,958,960],{"className":959},[103],[59,961,644],{"className":962},[103],[59,964,564],{"className":965},[563],[59,967,969],{"className":968},[170],[59,970,972],{"className":971,"style":710},[174],[59,973],{},[59,975,977],{"className":976,"style":509},[400,508],[59,978,487],{"className":979},[513,669],[59,981],{"className":982,"style":983},[870],"margin-right:0.1667em;",[59,985,987,993,1044],{"className":986},[504],[59,988,990],{"className":989,"style":509},[393,508],[59,991,455],{"className":992},[513,188],[59,994,996],{"className":995},[103],[59,997,999],{"className":998},[457],[59,1000,1002],{"className":1001},[523],[59,1003,1005,1036],{"className":1004},[166,527],[59,1006,1008,1033],{"className":1007},[170],[59,1009,1011,1022],{"className":1010,"style":534},[174],[59,1012,1013,1016],{"style":537},[59,1014],{"className":1015,"style":541},[181],[59,1017,1019],{"className":1018},[103],[59,1020,475],{"className":1021},[103],[59,1023,1024,1027],{"style":550},[59,1025],{"className":1026,"style":541},[181],[59,1028,1030],{"className":1029},[103],[59,1031,484],{"className":1032},[103],[59,1034,564],{"className":1035},[563],[59,1037,1039],{"className":1038},[170],[59,1040,1042],{"className":1041,"style":571},[174],[59,1043],{},[59,1045,1047],{"className":1046,"style":509},[400,508],[59,1048,487],{"className":1049},[513,188],[59,1051],{"className":1052,"style":871},[870],[59,1054,791],{"className":1055},[875],[59,1057],{"className":1058,"style":871},[870],[59,1060,1062,1066],{"className":1061},[94],[59,1063],{"className":1064,"style":1065},[98],"height:0.6444em;",[59,1067,842],{"className":1068},[103],[14,1070,1071],{},"所有行向量构成的空间，正是列向量空间的对偶空间。这听起来像是同一组数字换了个摆法——但当坐标系发生变换时，这两个对象的「命运」迥异，这正是问题的关键所在。",[14,1073,1074,1075,1111],{},"想象把单位从米换成厘米（这相当于把坐标基缩小为原来的 ",[59,1076,1078,1099],{"className":1077,"translate":63},[62],[59,1079,1081],{"className":1080},[67],[69,1082,1083],{"xmlns":71},[73,1084,1085,1096],{},[76,1086,1087,1089,1093],{},[473,1088,637],{},[79,1090,1092],{"mathvariant":1091},"normal","\u002F",[473,1094,1095],{},"100",[83,1097,1098],{"encoding":85},"1\u002F100",[59,1100,1102],{"className":1101,"ariaHidden":90},[89],[59,1103,1105,1108],{"className":1104},[94],[59,1106],{"className":1107,"style":386},[98],[59,1109,1098],{"className":1110},[103],"）。",[406,1113,1114,1117],{},[409,1115,1116],{},"以厘米度量时，向量分量被放大 100 倍（同一支箭头，单位更小，需要的刻度数更多）。",[409,1118,1119,1120,1152],{},"而对偶向量分量（即「每单位对应多少刻度」的标尺密度量）则缩小为 ",[59,1121,1123,1140],{"className":1122,"translate":63},[62],[59,1124,1126],{"className":1125},[67],[69,1127,1128],{"xmlns":71},[73,1129,1130,1138],{},[76,1131,1132,1134,1136],{},[473,1133,637],{},[79,1135,1092],{"mathvariant":1091},[473,1137,1095],{},[83,1139,1098],{"encoding":85},[59,1141,1143],{"className":1142,"ariaHidden":90},[89],[59,1144,1146,1149],{"className":1145},[94],[59,1147],{"className":1148,"style":386},[98],[59,1150,1098],{"className":1151},[103],"（刻度线本身变得更密；密度本身不变，但「每单位多少刻度」这个数字变了）。",[14,1154,1155,1156,1310],{},"两者相乘：",[59,1157,1159,1187],{"className":1158,"translate":63},[62],[59,1160,1162],{"className":1161},[67],[69,1163,1164],{"xmlns":71},[73,1165,1166,1184],{},[76,1167,1168,1170,1173,1180,1182],{},[473,1169,1095],{},[137,1171,1172],{},"×",[1174,1175,1176,1178],"mfrac",{},[473,1177,637],{},[473,1179,1095],{},[137,1181,791],{},[473,1183,637],{},[83,1185,1186],{"encoding":85},"100 \\times \\frac{1}{100} = 1",[59,1188,1190,1209,1301],{"className":1189,"ariaHidden":90},[89],[59,1191,1193,1197,1200,1203,1206],{"className":1192},[94],[59,1194],{"className":1195,"style":1196},[98],"height:0.7278em;vertical-align:-0.0833em;",[59,1198,1095],{"className":1199},[103],[59,1201],{"className":1202,"style":105},[870],[59,1204,1172],{"className":1205},[193],[59,1207],{"className":1208,"style":105},[870],[59,1210,1212,1216,1292,1295,1298],{"className":1211},[94],[59,1213],{"className":1214,"style":1215},[98],"height:1.1901em;vertical-align:-0.345em;",[59,1217,1219,1223,1289],{"className":1218},[103],[59,1220],{"className":1221},[393,1222],"nulldelimiter",[59,1224,1226],{"className":1225},[1174],[59,1227,1229,1280],{"className":1228},[166,527],[59,1230,1232,1277],{"className":1231},[170],[59,1233,1236,1251,1262],{"className":1234,"style":1235},[174],"height:0.8451em;",[59,1237,1239,1242],{"style":1238},"top:-2.655em;",[59,1240],{"className":1241,"style":541},[181],[59,1243,1245],{"className":1244},[186,187,188,189],[59,1246,1248],{"className":1247},[103,189],[59,1249,1095],{"className":1250},[103,189],[59,1252,1254,1257],{"style":1253},"top:-3.23em;",[59,1255],{"className":1256,"style":541},[181],[59,1258],{"className":1259,"style":1261},[1260],"frac-line","border-bottom-width:0.04em;",[59,1263,1265,1268],{"style":1264},"top:-3.394em;",[59,1266],{"className":1267,"style":541},[181],[59,1269,1271],{"className":1270},[186,187,188,189],[59,1272,1274],{"className":1273},[103,189],[59,1275,637],{"className":1276},[103,189],[59,1278,564],{"className":1279},[563],[59,1281,1283],{"className":1282},[170],[59,1284,1287],{"className":1285,"style":1286},[174],"height:0.345em;",[59,1288],{},[59,1290],{"className":1291},[400,1222],[59,1293],{"className":1294,"style":871},[870],[59,1296,791],{"className":1297},[875],[59,1299],{"className":1300,"style":871},[870],[59,1302,1304,1307],{"className":1303},[94],[59,1305],{"className":1306,"style":1065},[98],[59,1308,637],{"className":1309},[103],"，配对的结果——那个实数——保持不变。",[14,1312,1313],{},"这正是「逆变」（向量，其分量与基的变换方向相反）与「协变」（对偶向量，其分量与基的变换方向相同）这两个物理学名词的来历。对偶空间存在的全部理由，就是要保证最终算出的那个数——无论是长度、功、能量还是概率——是一个客观事实，不依赖于你选择米还是厘米，也不依赖于你采用哪套坐标系。",[14,1315,1316],{},"这也是全篇最重要的一句话：向量与对偶向量构成互补的一对；它们各自的分量随坐标选择而变化，但二者配对得到的标量永远不变。张量理论的每一项技术细节，归根结底都在为守护这个不变性服务。",[30,1318,1319],{"id":1319},"张量的严格定义",[14,1321,1322],{},"向量和对偶向量就位之后，张量的定义便自然浮现：",[11,1324,1325],{},[14,1326,1327,1382,1383,1413,1414,1443,1444,1472],{},[59,1328,1330,1354],{"className":1329,"translate":63},[62],[59,1331,1333],{"className":1332},[67],[69,1334,1335],{"xmlns":71},[73,1336,1337,1351],{},[76,1338,1339,1341,1343,1346,1349],{},[137,1340,368],{"stretchy":367},[79,1342,14],{},[137,1344,1345],{"separator":90},",",[79,1347,1348],{},"q",[137,1350,373],{"stretchy":367},[83,1352,1353],{"encoding":85},"(p,q)",[59,1355,1357],{"className":1356,"ariaHidden":90},[89],[59,1358,1360,1363,1366,1369,1373,1376,1379],{"className":1359},[94],[59,1361],{"className":1362,"style":386},[98],[59,1364,368],{"className":1365},[393],[59,1367,14],{"className":1368},[103,104],[59,1370,1345],{"className":1371},[1372],"mpunct",[59,1374],{"className":1375,"style":983},[870],[59,1377,1348],{"className":1378,"style":314},[103,104],[59,1380,373],{"className":1381},[400]," 型张量 ",[59,1384,1386,1400],{"className":1385,"translate":63},[62],[59,1387,1389],{"className":1388},[67],[69,1390,1391],{"xmlns":71},[73,1392,1393,1398],{},[76,1394,1395],{},[79,1396,1397],{},"T",[83,1399,1397],{"encoding":85},[59,1401,1403],{"className":1402,"ariaHidden":90},[89],[59,1404,1406,1409],{"className":1405},[94],[59,1407],{"className":1408,"style":99},[98],[59,1410,1397],{"className":1411,"style":1412},[103,104],"margin-right:0.1389em;"," 是一个多重线性映射：它同时接受 ",[59,1415,1417,1430],{"className":1416,"translate":63},[62],[59,1418,1420],{"className":1419},[67],[69,1421,1422],{"xmlns":71},[73,1423,1424,1428],{},[76,1425,1426],{},[79,1427,14],{},[83,1429,14],{"encoding":85},[59,1431,1433],{"className":1432,"ariaHidden":90},[89],[59,1434,1436,1440],{"className":1435},[94],[59,1437],{"className":1438,"style":1439},[98],"height:0.625em;vertical-align:-0.1944em;",[59,1441,14],{"className":1442},[103,104]," 个对偶向量和 ",[59,1445,1447,1460],{"className":1446,"translate":63},[62],[59,1448,1450],{"className":1449},[67],[69,1451,1452],{"xmlns":71},[73,1453,1454,1458],{},[76,1455,1456],{},[79,1457,1348],{},[83,1459,1348],{"encoding":85},[59,1461,1463],{"className":1462,"ariaHidden":90},[89],[59,1464,1466,1469],{"className":1465},[94],[59,1467],{"className":1468,"style":1439},[98],[59,1470,1348],{"className":1471,"style":314},[103,104]," 个向量作为参数，输出一个实数。",[14,1474,1475],{},"用符号表示：",[59,1477,1479],{"className":1478,"translate":63},[767],[59,1480,1482,1559],{"className":1481,"translate":63},[62],[59,1483,1485],{"className":1484},[67],[69,1486,1487],{"xmlns":71,"display":776},[73,1488,1489,1556],{},[76,1490,1491,1493,1496,1527,1529,1549,1552],{},[79,1492,1397],{},[137,1494,1495],{},":",[1497,1498,1499,1525],"munder",{},[1497,1500,1501,1522],{},[76,1502,1503,1509,1511,1514,1516],{},[132,1504,1505,1507],{},[79,1506,81],{},[137,1508,139],{},[137,1510,1172],{},[137,1512,1513],{},"⋯",[137,1515,1172],{},[132,1517,1518,1520],{},[79,1519,81],{},[137,1521,139],{},[137,1523,1524],{"stretchy":90},"⏟",[79,1526,14],{},[137,1528,1172],{},[1497,1530,1531,1547],{},[1497,1532,1533,1545],{},[76,1534,1535,1537,1539,1541,1543],{},[79,1536,81],{},[137,1538,1172],{},[137,1540,1513],{},[137,1542,1172],{},[79,1544,81],{},[137,1546,1524],{"stretchy":90},[79,1548,1348],{},[137,1550,1551],{},"⟶",[79,1553,1555],{"mathvariant":1554},"double-struck","R",[83,1557,1558],{"encoding":85},"T: \\underbrace{V^* \\times \\cdots \\times V^*}_{p} \\times \\underbrace{V \\times \\cdots \\times V}_{q} \\longrightarrow \\mathbb{R}",[59,1560,1562,1580,1804,1950],{"className":1561,"ariaHidden":90},[89],[59,1563,1565,1568,1571,1574,1577],{"className":1564},[94],[59,1566],{"className":1567,"style":99},[98],[59,1569,1397],{"className":1570,"style":1412},[103,104],[59,1572],{"className":1573,"style":871},[870],[59,1575,1495],{"className":1576},[875],[59,1578],{"className":1579,"style":871},[870],[59,1581,1583,1587,1795,1798,1801],{"className":1582},[94],[59,1584],{"className":1585,"style":1586},[98],"height:2.1075em;vertical-align:-1.3688em;",[59,1588,1590],{"className":1589},[504,1497],[59,1591,1593,1786],{"className":1592},[166,527],[59,1594,1596,1783],{"className":1595},[170],[59,1597,1600,1615],{"className":1598,"style":1599},[174],"height:0.7387em;",[59,1601,1603,1606],{"style":1602},"top:-1.7673em;",[59,1604],{"className":1605,"style":541},[181],[59,1607,1609],{"className":1608},[186,187,188,189],[59,1610,1612],{"className":1611},[103,189],[59,1613,14],{"className":1614},[103,104,189],[59,1616,1618,1621],{"style":1617},"top:-3em;",[59,1619],{"className":1620,"style":541},[181],[59,1622,1624],{"className":1623},[504,1497],[59,1625,1627,1774],{"className":1626},[166,527],[59,1628,1630,1771],{"className":1629},[170],[59,1631,1633,1683],{"className":1632,"style":1599},[174],[59,1634,1638,1641],{"className":1635,"style":1637},[1636],"svg-align","top:-2.2687em;",[59,1639],{"className":1640,"style":541},[181],[59,1642,1646,1663,1673],{"className":1643,"style":1645},[1644],"stretchy","height:0.548em;min-width:1.6em;",[59,1647,1651],{"className":1648,"style":1650},[1649],"brace-left","height:0.548em;",[1652,1653,1659],"svg",{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1658},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","0.548em","0 0 400000 548","xMinYMin slice",[1660,1661],"path",{"d":1662},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[59,1664,1667],{"className":1665,"style":1650},[1666],"brace-center",[1652,1668,1670],{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1669},"xMidYMin slice",[1660,1671],{"d":1672},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[59,1674,1677],{"className":1675,"style":1650},[1676],"brace-right",[1652,1678,1680],{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1679},"xMaxYMin slice",[1660,1681],{"d":1682},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[59,1684,1685,1688],{"style":1617},[59,1686],{"className":1687,"style":541},[181],[59,1689,1691,1721,1724,1727,1730,1733,1736,1739,1742],{"className":1690},[103],[59,1692,1694,1697],{"className":1693},[103],[59,1695,81],{"className":1696,"style":105},[103,104],[59,1698,1700],{"className":1699},[162],[59,1701,1703],{"className":1702},[166],[59,1704,1706],{"className":1705},[170],[59,1707,1709],{"className":1708,"style":1599},[174],[59,1710,1712,1715],{"style":1711},"top:-3.113em;margin-right:0.05em;",[59,1713],{"className":1714,"style":182},[181],[59,1716,1718],{"className":1717},[186,187,188,189],[59,1719,139],{"className":1720},[193,189],[59,1722],{"className":1723,"style":105},[870],[59,1725,1172],{"className":1726},[193],[59,1728],{"className":1729,"style":105},[870],[59,1731,1513],{"className":1732},[504],[59,1734],{"className":1735,"style":105},[870],[59,1737,1172],{"className":1738},[193],[59,1740],{"className":1741,"style":105},[870],[59,1743,1745,1748],{"className":1744},[103],[59,1746,81],{"className":1747,"style":105},[103,104],[59,1749,1751],{"className":1750},[162],[59,1752,1754],{"className":1753},[166],[59,1755,1757],{"className":1756},[170],[59,1758,1760],{"className":1759,"style":1599},[174],[59,1761,1762,1765],{"style":1711},[59,1763],{"className":1764,"style":182},[181],[59,1766,1768],{"className":1767},[186,187,188,189],[59,1769,139],{"className":1770},[193,189],[59,1772,564],{"className":1773},[563],[59,1775,1777],{"className":1776},[170],[59,1778,1781],{"className":1779,"style":1780},[174],"height:0.7313em;",[59,1782],{},[59,1784,564],{"className":1785},[563],[59,1787,1789],{"className":1788},[170],[59,1790,1793],{"className":1791,"style":1792},[174],"height:1.3688em;",[59,1794],{},[59,1796],{"className":1797,"style":105},[870],[59,1799,1172],{"className":1800},[193],[59,1802],{"className":1803,"style":105},[870],[59,1805,1807,1811,1941,1944,1947],{"className":1806},[94],[59,1808],{"className":1809,"style":1810},[98],"height:2.0522em;vertical-align:-1.3688em;",[59,1812,1814],{"className":1813},[504,1497],[59,1815,1817,1933],{"className":1816},[166,527],[59,1818,1820,1930],{"className":1819},[170],[59,1821,1823,1837],{"className":1822,"style":99},[174],[59,1824,1825,1828],{"style":1602},[59,1826],{"className":1827,"style":541},[181],[59,1829,1831],{"className":1830},[186,187,188,189],[59,1832,1834],{"className":1833},[103,189],[59,1835,1348],{"className":1836,"style":314},[103,104,189],[59,1838,1839,1842],{"style":1617},[59,1840],{"className":1841,"style":541},[181],[59,1843,1845],{"className":1844},[504,1497],[59,1846,1848,1922],{"className":1847},[166,527],[59,1849,1851,1919],{"className":1850},[170],[59,1852,1854,1884],{"className":1853,"style":99},[174],[59,1855,1857,1860],{"className":1856,"style":1637},[1636],[59,1858],{"className":1859,"style":541},[181],[59,1861,1863,1870,1877],{"className":1862,"style":1645},[1644],[59,1864,1866],{"className":1865,"style":1650},[1649],[1652,1867,1868],{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1658},[1660,1869],{"d":1662},[59,1871,1873],{"className":1872,"style":1650},[1666],[1652,1874,1875],{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1669},[1660,1876],{"d":1672},[59,1878,1880],{"className":1879,"style":1650},[1676],[1652,1881,1882],{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1679},[1660,1883],{"d":1682},[59,1885,1886,1889],{"style":1617},[59,1887],{"className":1888,"style":541},[181],[59,1890,1892,1895,1898,1901,1904,1907,1910,1913,1916],{"className":1891},[103],[59,1893,81],{"className":1894,"style":105},[103,104],[59,1896],{"className":1897,"style":105},[870],[59,1899,1172],{"className":1900},[193],[59,1902],{"className":1903,"style":105},[870],[59,1905,1513],{"className":1906},[504],[59,1908],{"className":1909,"style":105},[870],[59,1911,1172],{"className":1912},[193],[59,1914],{"className":1915,"style":105},[870],[59,1917,81],{"className":1918,"style":105},[103,104],[59,1920,564],{"className":1921},[563],[59,1923,1925],{"className":1924},[170],[59,1926,1928],{"className":1927,"style":1780},[174],[59,1929],{},[59,1931,564],{"className":1932},[563],[59,1934,1936],{"className":1935},[170],[59,1937,1939],{"className":1938,"style":1792},[174],[59,1940],{},[59,1942],{"className":1943,"style":871},[870],[59,1945,1551],{"className":1946},[875],[59,1948],{"className":1949,"style":871},[870],[59,1951,1953,1957],{"className":1952},[94],[59,1954],{"className":1955,"style":1956},[98],"height:0.6889em;",[59,1958,1555],{"className":1959},[103,1960],"mathbb",[14,1962,1963],{},"两个关键词，一个都不能少：",[406,1965,1966,1969],{},[409,1967,1968],{},"多重线性：对每个槽位分别线性（加法保持、对标量乘法齐次）。这保证张量运算可以像矩阵运算一样，完全展开为分量的加权和。",[409,1970,1971],{},"标量输出：无论有多少个槽位、输入的是什么类型，最终出来的都是一个不随坐标变化而变化的绝对数字。",[14,1973,1974],{},"由此直接得到张量阶数的公式：",[59,1976,1978],{"className":1977,"translate":63},[767],[59,1979,1981,2006],{"className":1980,"translate":63},[62],[59,1982,1984],{"className":1983},[67],[69,1985,1986],{"xmlns":71,"display":776},[73,1987,1988,2003],{},[76,1989,1990,1994,1996,1998,2001],{},[1991,1992,1993],"mtext",{},"阶数",[137,1995,791],{},[79,1997,14],{},[137,1999,2000],{},"+",[79,2002,1348],{},[83,2004,2005],{"encoding":85},"\\text{阶数} = p + q",[59,2007,2009,2032,2051],{"className":2008,"ariaHidden":90},[89],[59,2010,2012,2015,2023,2026,2029],{"className":2011},[94],[59,2013],{"className":2014,"style":99},[98],[59,2016,2019],{"className":2017},[103,2018],"text",[59,2020,1993],{"className":2021},[103,2022],"cjk_fallback",[59,2024],{"className":2025,"style":871},[870],[59,2027,791],{"className":2028},[875],[59,2030],{"className":2031,"style":871},[870],[59,2033,2035,2039,2042,2045,2048],{"className":2034},[94],[59,2036],{"className":2037,"style":2038},[98],"height:0.7778em;vertical-align:-0.1944em;",[59,2040,14],{"className":2041},[103,104],[59,2043],{"className":2044,"style":105},[870],[59,2046,2000],{"className":2047},[193],[59,2049],{"className":2050,"style":105},[870],[59,2052,2054,2057],{"className":2053},[94],[59,2055],{"className":2056,"style":1439},[98],[59,2058,1348],{"className":2059,"style":314},[103,104],[14,2061,2062],{},"这也是为什么单说「阶数」信息量不够——它只告诉你张量有几个槽位，却不告诉你哪些槽位吃向量、哪些槽位吃对偶向量。这种区分绝非咬文嚼字；下一章将用一个具体例子说明，为什么它是生死攸关的问题。",[30,2064,2065],{"id":2065},"槽位哲学",[14,2067,2068],{},"物理教科书常常这样描述二阶张量：",[11,2070,2071],{},[14,2072,2073,2074,2104,2105,2135,2136,2167],{},"「应力张量 ",[59,2075,2077,2092],{"className":2076,"translate":63},[62],[59,2078,2080],{"className":2079},[67],[69,2081,2082],{"xmlns":71},[73,2083,2084,2089],{},[76,2085,2086],{},[79,2087,2088],{},"σ",[83,2090,2091],{"encoding":85},"\\sigma",[59,2093,2095],{"className":2094,"ariaHidden":90},[89],[59,2096,2098,2101],{"className":2097},[94],[59,2099],{"className":2100,"style":310},[98],[59,2102,2088],{"className":2103,"style":314},[103,104]," 作用在一个法向量 ",[59,2106,2108,2123],{"className":2107,"translate":63},[62],[59,2109,2111],{"className":2110},[67],[69,2112,2113],{"xmlns":71},[73,2114,2115,2120],{},[76,2116,2117],{},[79,2118,2119],{"mathvariant":330},"n",[83,2121,2122],{"encoding":85},"\\mathbf{n}",[59,2124,2126],{"className":2125,"ariaHidden":90},[89],[59,2127,2129,2132],{"className":2128},[94],[59,2130],{"className":2131,"style":344},[98],[59,2133,2119],{"className":2134},[103,348]," 上，产生一个应力向量 ",[59,2137,2139,2154],{"className":2138,"translate":63},[62],[59,2140,2142],{"className":2141},[67],[69,2143,2144],{"xmlns":71},[73,2145,2146,2151],{},[76,2147,2148],{},[79,2149,2150],{"mathvariant":330},"t",[83,2152,2153],{"encoding":85},"\\mathbf{t}",[59,2155,2157],{"className":2156,"ariaHidden":90},[89],[59,2158,2160,2164],{"className":2159},[94],[59,2161],{"className":2162,"style":2163},[98],"height:0.6349em;",[59,2165,2150],{"className":2166},[103,348],"。」",[14,2169,2170],{},"这句话听起来像是「输入一个向量，输出一个向量」。而第 3 章的定义明明说，张量是「输入几个向量和对偶向量、输出一个标量」的东西。两种说法似乎完全不可调和——而这正是绝大多数读者卡壳的地方。",[14,2172,2173],{},"解开这个矛盾的钥匙是：多重线性映射并不要求你一次填满所有槽位。",[14,2175,2176,2177,194,2205,2257,2258,2286],{},"应力张量 ",[59,2178,2180,2193],{"className":2179,"translate":63},[62],[59,2181,2183],{"className":2182},[67],[69,2184,2185],{"xmlns":71},[73,2186,2187,2191],{},[76,2188,2189],{},[79,2190,2088],{},[83,2192,2091],{"encoding":85},[59,2194,2196],{"className":2195,"ariaHidden":90},[89],[59,2197,2199,2202],{"className":2198},[94],[59,2200],{"className":2201,"style":310},[98],[59,2203,2088],{"className":2204,"style":314},[103,104],[59,2206,2208,2230],{"className":2207,"translate":63},[62],[59,2209,2211],{"className":2210},[67],[69,2212,2213],{"xmlns":71},[73,2214,2215,2227],{},[76,2216,2217,2219,2221,2223,2225],{},[137,2218,368],{"stretchy":367},[473,2220,637],{},[137,2222,1345],{"separator":90},[473,2224,637],{},[137,2226,373],{"stretchy":367},[83,2228,2229],{"encoding":85},"(1,1)",[59,2231,2233],{"className":2232,"ariaHidden":90},[89],[59,2234,2236,2239,2242,2245,2248,2251,2254],{"className":2235},[94],[59,2237],{"className":2238,"style":386},[98],[59,2240,368],{"className":2241},[393],[59,2243,637],{"className":2244},[103],[59,2246,1345],{"className":2247},[1372],[59,2249],{"className":2250,"style":983},[870],[59,2252,637],{"className":2253},[103],[59,2255,373],{"className":2256},[400]," 型张量，有两个槽位：一个对偶向量槽位和一个向量槽位。当你只把法向量 ",[59,2259,2261,2274],{"className":2260,"translate":63},[62],[59,2262,2264],{"className":2263},[67],[69,2265,2266],{"xmlns":71},[73,2267,2268,2272],{},[76,2269,2270],{},[79,2271,2119],{"mathvariant":330},[83,2273,2122],{"encoding":85},[59,2275,2277],{"className":2276,"ariaHidden":90},[89],[59,2278,2280,2283],{"className":2279},[94],[59,2281],{"className":2282,"style":344},[98],[59,2284,2119],{"className":2285},[103,348]," 塞进它的向量槽位时：",[406,2288,2289,2292],{},[409,2290,2291],{},"向量槽位已被填满；",[409,2293,2294],{},"对偶向量槽位仍然空着。",[14,2296,2297,2298,2356,2357,1111],{},"此时 ",[59,2299,2301,2326],{"className":2300,"translate":63},[62],[59,2302,2304],{"className":2303},[67],[69,2305,2306],{"xmlns":71},[73,2307,2308,2323],{},[76,2309,2310,2312,2314,2317,2319,2321],{},[79,2311,2088],{},[137,2313,368],{"stretchy":367},[137,2315,2316],{},"⋅",[137,2318,1345],{"separator":90},[79,2320,2119],{"mathvariant":330},[137,2322,373],{"stretchy":367},[83,2324,2325],{"encoding":85},"\\sigma(\\cdot, \\mathbf{n})",[59,2327,2329],{"className":2328,"ariaHidden":90},[89],[59,2330,2332,2335,2338,2341,2344,2347,2350,2353],{"className":2331},[94],[59,2333],{"className":2334,"style":386},[98],[59,2336,2088],{"className":2337,"style":314},[103,104],[59,2339,368],{"className":2340},[393],[59,2342,2316],{"className":2343},[103],[59,2345,1345],{"className":2346},[1372],[59,2348],{"className":2349,"style":983},[870],[59,2351,2119],{"className":2352},[103,348],[59,2354,373],{"className":2355},[400]," 变成了什么？它变成了一个仍在等待对偶向量输入的中途产物——而「一台等待对偶向量、随后输出标量的机器」，恰好是定义向量的另一种严格方式（通过自然同构 ",[59,2358,2360,2387],{"className":2359,"translate":63},[62],[59,2361,2363],{"className":2362},[67],[69,2364,2365],{"xmlns":71},[73,2366,2367,2384],{},[76,2368,2369,2371,2374],{},[79,2370,81],{},[137,2372,2373],{},"≅",[132,2375,2376,2378],{},[79,2377,81],{},[76,2379,2380,2382],{},[137,2381,139],{},[137,2383,139],{},[83,2385,2386],{"encoding":85},"V \\cong V^{**}",[59,2388,2390,2408],{"className":2389,"ariaHidden":90},[89],[59,2391,2393,2396,2399,2402,2405],{"className":2392},[94],[59,2394],{"className":2395,"style":99},[98],[59,2397,81],{"className":2398,"style":105},[103,104],[59,2400],{"className":2401,"style":871},[870],[59,2403,2373],{"className":2404},[875],[59,2406],{"className":2407,"style":871},[870],[59,2409,2411,2414],{"className":2410},[94],[59,2412],{"className":2413,"style":152},[98],[59,2415,2417,2420],{"className":2416},[103],[59,2418,81],{"className":2419,"style":105},[103,104],[59,2421,2423],{"className":2422},[162],[59,2424,2426],{"className":2425},[166],[59,2427,2429],{"className":2428},[170],[59,2430,2432],{"className":2431,"style":152},[174],[59,2433,2434,2437],{"style":177},[59,2435],{"className":2436,"style":182},[181],[59,2438,2440],{"className":2439},[186,187,188,189],[59,2441,2443],{"className":2442},[103,189],[59,2444,2446],{"className":2445},[103,189],"∗∗",[406,2448,2449,2510],{},[409,2450,2451,2452,2480,2481,2509],{},"只填一个槽位：",[59,2453,2455,2468],{"className":2454,"translate":63},[62],[59,2456,2458],{"className":2457},[67],[69,2459,2460],{"xmlns":71},[73,2461,2462,2466],{},[76,2463,2464],{},[79,2465,2088],{},[83,2467,2091],{"encoding":85},[59,2469,2471],{"className":2470,"ariaHidden":90},[89],[59,2472,2474,2477],{"className":2473},[94],[59,2475],{"className":2476,"style":310},[98],[59,2478,2088],{"className":2479,"style":314},[103,104]," 退化成一个「待定」向量，这正是教科书里说的「输出应力向量 ",[59,2482,2484,2497],{"className":2483,"translate":63},[62],[59,2485,2487],{"className":2486},[67],[69,2488,2489],{"xmlns":71},[73,2490,2491,2495],{},[76,2492,2493],{},[79,2494,2150],{"mathvariant":330},[83,2496,2153],{"encoding":85},[59,2498,2500],{"className":2499,"ariaHidden":90},[89],[59,2501,2503,2506],{"className":2502},[94],[59,2504],{"className":2505,"style":2163},[98],[59,2507,2150],{"className":2508},[103,348],"」；",[409,2511,2512],{},"两个槽位都填满（法向量 + 某个方向上的对偶向量\u002F标尺）：张量终于输出一个标量，即该方向上应力分量的大小。",[14,2514,2515],{},"因此，「输入一个向量、输出一个向量」与「输入两个参数、输出一个标量」并不矛盾；前者只是后者的中间状态。物理教科书为了照顾直觉，通常只展示「喂了一半」的结果；数学定义则严谨地描述所有槽位填满后的最终行为。理解了这一层，关于张量「到底是映射还是运算」的争论几乎会自行消解。",[14,2517,2518,2519,2547,2548,2576],{},"那么，为什么必须区分 ",[59,2520,2522,2535],{"className":2521,"translate":63},[62],[59,2523,2525],{"className":2524},[67],[69,2526,2527],{"xmlns":71},[73,2528,2529,2533],{},[76,2530,2531],{},[79,2532,14],{},[83,2534,14],{"encoding":85},[59,2536,2538],{"className":2537,"ariaHidden":90},[89],[59,2539,2541,2544],{"className":2540},[94],[59,2542],{"className":2543,"style":1439},[98],[59,2545,14],{"className":2546},[103,104]," 和 ",[59,2549,2551,2564],{"className":2550,"translate":63},[62],[59,2552,2554],{"className":2553},[67],[69,2555,2556],{"xmlns":71},[73,2557,2558,2562],{},[76,2559,2560],{},[79,2561,1348],{},[83,2563,1348],{"encoding":85},[59,2565,2567],{"className":2566,"ariaHidden":90},[89],[59,2568,2570,2573],{"className":2569},[94],[59,2571],{"className":2572,"style":1439},[98],[59,2574,1348],{"className":2575,"style":314},[103,104],"，而不是只说阶数？",[14,2578,2579,2630,2631,2683],{},[59,2580,2582,2603],{"className":2581,"translate":63},[62],[59,2583,2585],{"className":2584},[67],[69,2586,2587],{"xmlns":71},[73,2588,2589,2601],{},[76,2590,2591,2593,2595,2597,2599],{},[137,2592,368],{"stretchy":367},[473,2594,637],{},[137,2596,1345],{"separator":90},[473,2598,637],{},[137,2600,373],{"stretchy":367},[83,2602,2229],{"encoding":85},[59,2604,2606],{"className":2605,"ariaHidden":90},[89],[59,2607,2609,2612,2615,2618,2621,2624,2627],{"className":2608},[94],[59,2610],{"className":2611,"style":386},[98],[59,2613,368],{"className":2614},[393],[59,2616,637],{"className":2617},[103],[59,2619,1345],{"className":2620},[1372],[59,2622],{"className":2623,"style":983},[870],[59,2625,637],{"className":2626},[103],[59,2628,373],{"className":2629},[400]," 型张量与 ",[59,2632,2634,2656],{"className":2633,"translate":63},[62],[59,2635,2637],{"className":2636},[67],[69,2638,2639],{"xmlns":71},[73,2640,2641,2653],{},[76,2642,2643,2645,2647,2649,2651],{},[137,2644,368],{"stretchy":367},[473,2646,471],{},[137,2648,1345],{"separator":90},[473,2650,475],{},[137,2652,373],{"stretchy":367},[83,2654,2655],{"encoding":85},"(0,2)",[59,2657,2659],{"className":2658,"ariaHidden":90},[89],[59,2660,2662,2665,2668,2671,2674,2677,2680],{"className":2661},[94],[59,2663],{"className":2664,"style":386},[98],[59,2666,368],{"className":2667},[393],[59,2669,471],{"className":2670},[103],[59,2672,1345],{"className":2673},[1372],[59,2675],{"className":2676,"style":983},[870],[59,2678,475],{"className":2679},[103],[59,2681,373],{"className":2682},[400]," 型张量，写成数组时都是二维矩阵——视觉上无法区分。但它们在坐标变换下的行为截然不同：",[406,2685,2686,2740],{},[409,2687,2688,2739],{},[59,2689,2691,2712],{"className":2690,"translate":63},[62],[59,2692,2694],{"className":2693},[67],[69,2695,2696],{"xmlns":71},[73,2697,2698,2710],{},[76,2699,2700,2702,2704,2706,2708],{},[137,2701,368],{"stretchy":367},[473,2703,637],{},[137,2705,1345],{"separator":90},[473,2707,637],{},[137,2709,373],{"stretchy":367},[83,2711,2229],{"encoding":85},[59,2713,2715],{"className":2714,"ariaHidden":90},[89],[59,2716,2718,2721,2724,2727,2730,2733,2736],{"className":2717},[94],[59,2719],{"className":2720,"style":386},[98],[59,2722,368],{"className":2723},[393],[59,2725,637],{"className":2726},[103],[59,2728,1345],{"className":2729},[1372],[59,2731],{"className":2732,"style":983},[870],[59,2734,637],{"className":2735},[103],[59,2737,373],{"className":2738},[400]," 型（例如线性变换、惯性张量）：一个上标一个下标；变换时一个分量「逆变」、一个分量「协变」，两者相互抵消。",[409,2741,2742,2793],{},[59,2743,2745,2766],{"className":2744,"translate":63},[62],[59,2746,2748],{"className":2747},[67],[69,2749,2750],{"xmlns":71},[73,2751,2752,2764],{},[76,2753,2754,2756,2758,2760,2762],{},[137,2755,368],{"stretchy":367},[473,2757,471],{},[137,2759,1345],{"separator":90},[473,2761,475],{},[137,2763,373],{"stretchy":367},[83,2765,2655],{"encoding":85},[59,2767,2769],{"className":2768,"ariaHidden":90},[89],[59,2770,2772,2775,2778,2781,2784,2787,2790],{"className":2771},[94],[59,2773],{"className":2774,"style":386},[98],[59,2776,368],{"className":2777},[393],[59,2779,471],{"className":2780},[103],[59,2782,1345],{"className":2783},[1372],[59,2785],{"className":2786,"style":983},[870],[59,2788,475],{"className":2789},[103],[59,2791,373],{"className":2792},[400]," 型（例如度量张量）：两个下标；变换时两个分量都「协变」，向同一方向累积。",[14,2795,2796,2797,2849,2850,2901],{},"如果只说「阶数」而忽略 ",[59,2798,2800,2822],{"className":2799,"translate":63},[62],[59,2801,2803],{"className":2802},[67],[69,2804,2805],{"xmlns":71},[73,2806,2807,2819],{},[76,2808,2809,2811,2813,2815,2817],{},[137,2810,368],{"stretchy":367},[79,2812,14],{},[137,2814,1345],{"separator":90},[79,2816,1348],{},[137,2818,373],{"stretchy":367},[83,2820,2821],{"encoding":85},"(p, q)",[59,2823,2825],{"className":2824,"ariaHidden":90},[89],[59,2826,2828,2831,2834,2837,2840,2843,2846],{"className":2827},[94],[59,2829],{"className":2830,"style":386},[98],[59,2832,368],{"className":2833},[393],[59,2835,14],{"className":2836},[103,104],[59,2838,1345],{"className":2839},[1372],[59,2841],{"className":2842,"style":983},[870],[59,2844,1348],{"className":2845,"style":314},[103,104],[59,2847,373],{"className":2848},[400]," 之分，在计算散度、缩并等运算时必然误套公式——而在广义相对论里，这种错误足以让「能量守恒」这样的物理定律在计算层面崩塌。这就是数学家坚持用 ",[59,2851,2853,2874],{"className":2852,"translate":63},[62],[59,2854,2856],{"className":2855},[67],[69,2857,2858],{"xmlns":71},[73,2859,2860,2872],{},[76,2861,2862,2864,2866,2868,2870],{},[137,2863,368],{"stretchy":367},[79,2865,14],{},[137,2867,1345],{"separator":90},[79,2869,1348],{},[137,2871,373],{"stretchy":367},[83,2873,1353],{"encoding":85},[59,2875,2877],{"className":2876,"ariaHidden":90},[89],[59,2878,2880,2883,2886,2889,2892,2895,2898],{"className":2879},[94],[59,2881],{"className":2882,"style":386},[98],[59,2884,368],{"className":2885},[393],[59,2887,14],{"className":2888},[103,104],[59,2890,1345],{"className":2891},[1372],[59,2893],{"className":2894,"style":983},[870],[59,2896,1348],{"className":2897,"style":314},[103,104],[59,2899,373],{"className":2900},[400]," 型而非笼统的「阶数」来标注张量的原因。",[30,2903,2904],{"id":2904},"度量张量",[14,2906,2907],{},"至此还有一个问题悬而未决：向量和对偶向量居于两个显然不同的空间，物理学家却经常把二者混用（比如说「把一个向量的指标降下来」）。这到底是怎么回事？",[14,2909,2910,2911,2998],{},"在一个配有度量张量 ",[59,2912,2914,2940],{"className":2913,"translate":63},[62],[59,2915,2917],{"className":2916},[67],[69,2918,2919],{"xmlns":71},[73,2920,2921,2937],{},[76,2922,2923],{},[2924,2925,2926,2929],"msub",{},[79,2927,2928],{},"g",[76,2930,2931,2934],{},[79,2932,2933],{},"i",[79,2935,2936],{},"j",[83,2938,2939],{"encoding":85},"g_{ij}",[59,2941,2943],{"className":2942,"ariaHidden":90},[89],[59,2944,2946,2950],{"className":2945},[94],[59,2947],{"className":2948,"style":2949},[98],"height:0.7167em;vertical-align:-0.2861em;",[59,2951,2953,2956],{"className":2952},[103],[59,2954,2928],{"className":2955,"style":314},[103,104],[59,2957,2959],{"className":2958},[162],[59,2960,2962,2989],{"className":2961},[166,527],[59,2963,2965,2986],{"className":2964},[170],[59,2966,2969],{"className":2967,"style":2968},[174],"height:0.3117em;",[59,2970,2972,2975],{"style":2971},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[59,2973],{"className":2974,"style":182},[181],[59,2976,2978],{"className":2977},[186,187,188,189],[59,2979,2981],{"className":2980},[103,189],[59,2982,2985],{"className":2983,"style":2984},[103,104,189],"margin-right:0.0572em;","ij",[59,2987,564],{"className":2988},[563],[59,2990,2992],{"className":2991},[170],[59,2993,2996],{"className":2994,"style":2995},[174],"height:0.2861em;",[59,2997],{}," 的空间里（几乎所有的物理空间都是如此），存在一台自然的「翻译机」，能把向量翻译成对偶向量，也能反向翻译。这个操作叫作指标升降：",[14,3000,3001],{},"降指标：向量到对偶向量",[59,3003,3005],{"className":3004,"translate":63},[767],[59,3006,3008,3044],{"className":3007,"translate":63},[62],[59,3009,3011],{"className":3010},[67],[69,3012,3013],{"xmlns":71,"display":776},[73,3014,3015,3041],{},[76,3016,3017,3023,3025,3035],{},[2924,3018,3019,3021],{},[79,3020,331],{},[79,3022,2933],{},[137,3024,791],{},[2924,3026,3027,3029],{},[79,3028,2928],{},[76,3030,3031,3033],{},[79,3032,2933],{},[79,3034,2936],{},[132,3036,3037,3039],{},[79,3038,331],{},[79,3040,2936],{},[83,3042,3043],{"encoding":85},"v_i = g_{ij} v^j",[59,3045,3047,3104],{"className":3046,"ariaHidden":90},[89],[59,3048,3050,3054,3095,3098,3101],{"className":3049},[94],[59,3051],{"className":3052,"style":3053},[98],"height:0.5806em;vertical-align:-0.15em;",[59,3055,3057,3060],{"className":3056},[103],[59,3058,331],{"className":3059,"style":314},[103,104],[59,3061,3063],{"className":3062},[162],[59,3064,3066,3086],{"className":3065},[166,527],[59,3067,3069,3083],{"className":3068},[170],[59,3070,3072],{"className":3071,"style":2968},[174],[59,3073,3074,3077],{"style":2971},[59,3075],{"className":3076,"style":182},[181],[59,3078,3080],{"className":3079},[186,187,188,189],[59,3081,2933],{"className":3082},[103,104,189],[59,3084,564],{"className":3085},[563],[59,3087,3089],{"className":3088},[170],[59,3090,3093],{"className":3091,"style":3092},[174],"height:0.15em;",[59,3094],{},[59,3096],{"className":3097,"style":871},[870],[59,3099,791],{"className":3100},[875],[59,3102],{"className":3103,"style":871},[870],[59,3105,3107,3111,3154],{"className":3106},[94],[59,3108],{"className":3109,"style":3110},[98],"height:1.1608em;vertical-align:-0.2861em;",[59,3112,3114,3117],{"className":3113},[103],[59,3115,2928],{"className":3116,"style":314},[103,104],[59,3118,3120],{"className":3119},[162],[59,3121,3123,3146],{"className":3122},[166,527],[59,3124,3126,3143],{"className":3125},[170],[59,3127,3129],{"className":3128,"style":2968},[174],[59,3130,3131,3134],{"style":2971},[59,3132],{"className":3133,"style":182},[181],[59,3135,3137],{"className":3136},[186,187,188,189],[59,3138,3140],{"className":3139},[103,189],[59,3141,2985],{"className":3142,"style":2984},[103,104,189],[59,3144,564],{"className":3145},[563],[59,3147,3149],{"className":3148},[170],[59,3150,3152],{"className":3151,"style":2995},[174],[59,3153],{},[59,3155,3157,3160],{"className":3156},[103],[59,3158,331],{"className":3159,"style":314},[103,104],[59,3161,3163],{"className":3162},[162],[59,3164,3166],{"className":3165},[166],[59,3167,3169],{"className":3168},[170],[59,3170,3173],{"className":3171,"style":3172},[174],"height:0.8747em;",[59,3174,3175,3178],{"style":1711},[59,3176],{"className":3177,"style":182},[181],[59,3179,3181],{"className":3180},[186,187,188,189],[59,3182,2936],{"className":3183,"style":2984},[103,104,189],[14,3185,3186],{},"升指标：对偶向量到向量",[59,3188,3190],{"className":3189,"translate":63},[767],[59,3191,3193,3229],{"className":3192,"translate":63},[62],[59,3194,3196],{"className":3195},[67],[69,3197,3198],{"xmlns":71,"display":776},[73,3199,3200,3226],{},[76,3201,3202,3208,3210,3220],{},[132,3203,3204,3206],{},[79,3205,331],{},[79,3207,2933],{},[137,3209,791],{},[132,3211,3212,3214],{},[79,3213,2928],{},[76,3215,3216,3218],{},[79,3217,2933],{},[79,3219,2936],{},[2924,3221,3222,3224],{},[79,3223,331],{},[79,3225,2936],{},[83,3227,3228],{"encoding":85},"v^i = g^{ij} v_j",[59,3230,3232,3276],{"className":3231,"ariaHidden":90},[89],[59,3233,3235,3238,3267,3270,3273],{"className":3234},[94],[59,3236],{"className":3237,"style":3172},[98],[59,3239,3241,3244],{"className":3240},[103],[59,3242,331],{"className":3243,"style":314},[103,104],[59,3245,3247],{"className":3246},[162],[59,3248,3250],{"className":3249},[166],[59,3251,3253],{"className":3252},[170],[59,3254,3256],{"className":3255,"style":3172},[174],[59,3257,3258,3261],{"style":1711},[59,3259],{"className":3260,"style":182},[181],[59,3262,3264],{"className":3263},[186,187,188,189],[59,3265,2933],{"className":3266},[103,104,189],[59,3268],{"className":3269,"style":871},[870],[59,3271,791],{"className":3272},[875],[59,3274],{"className":3275,"style":871},[870],[59,3277,3279,3282,3314],{"className":3278},[94],[59,3280],{"className":3281,"style":3110},[98],[59,3283,3285,3288],{"className":3284},[103],[59,3286,2928],{"className":3287,"style":314},[103,104],[59,3289,3291],{"className":3290},[162],[59,3292,3294],{"className":3293},[166],[59,3295,3297],{"className":3296},[170],[59,3298,3300],{"className":3299,"style":3172},[174],[59,3301,3302,3305],{"style":1711},[59,3303],{"className":3304,"style":182},[181],[59,3306,3308],{"className":3307},[186,187,188,189],[59,3309,3311],{"className":3310},[103,189],[59,3312,2985],{"className":3313,"style":2984},[103,104,189],[59,3315,3317,3320],{"className":3316},[103],[59,3318,331],{"className":3319,"style":314},[103,104],[59,3321,3323],{"className":3322},[162],[59,3324,3326,3346],{"className":3325},[166,527],[59,3327,3329,3343],{"className":3328},[170],[59,3330,3332],{"className":3331,"style":2968},[174],[59,3333,3334,3337],{"style":2971},[59,3335],{"className":3336,"style":182},[181],[59,3338,3340],{"className":3339},[186,187,188,189],[59,3341,2936],{"className":3342,"style":2984},[103,104,189],[59,3344,564],{"className":3345},[563],[59,3347,3349],{"className":3348},[170],[59,3350,3352],{"className":3351,"style":2995},[174],[59,3353],{},[14,3355,3356,3357,3408],{},"度量张量本身是 ",[59,3358,3360,3381],{"className":3359,"translate":63},[62],[59,3361,3363],{"className":3362},[67],[69,3364,3365],{"xmlns":71},[73,3366,3367,3379],{},[76,3368,3369,3371,3373,3375,3377],{},[137,3370,368],{"stretchy":367},[473,3372,471],{},[137,3374,1345],{"separator":90},[473,3376,475],{},[137,3378,373],{"stretchy":367},[83,3380,2655],{"encoding":85},[59,3382,3384],{"className":3383,"ariaHidden":90},[89],[59,3385,3387,3390,3393,3396,3399,3402,3405],{"className":3386},[94],[59,3388],{"className":3389,"style":386},[98],[59,3391,368],{"className":3392},[393],[59,3394,471],{"className":3395},[103],[59,3397,1345],{"className":3398},[1372],[59,3400],{"className":3401,"style":983},[870],[59,3403,475],{"className":3404},[103],[59,3406,373],{"className":3407},[400]," 型张量；它的作用正是「吞进两个向量，输出它们的内积」——而内积正是长度与角度的定义。正因有了度量，我们才能谈论「向量的长度」这个听起来像是向量固有属性、实际上离开对偶配对就无法定义的量。",[14,3410,3411],{},"在欧几里得空间（比如熟悉的笛卡尔坐标系）里，度量恰好是单位矩阵，指标升降「看起来」什么也没做。这也是为什么中学物理里向量与对偶向量的区别被完全掩盖了。一旦坐标系变得复杂（极坐标、球坐标），或者空间本身弯曲（广义相对论的时空），度量就不再是单位矩阵，向量与对偶向量的区别就会以决定性的姿态重新登场。",[30,3413,3414],{"id":3414},"坐标变换法则",[14,3416,3417,3418,3447,3448,47,3513,3564],{},"前面几章都在诉诸直觉；这里给出具体的分量变换法则作为「锚」。设坐标从 ",[59,3419,3421,3435],{"className":3420,"translate":63},[62],[59,3422,3424],{"className":3423},[67],[69,3425,3426],{"xmlns":71},[73,3427,3428,3433],{},[76,3429,3430],{},[79,3431,3432],{},"x",[83,3434,3432],{"encoding":85},[59,3436,3438],{"className":3437,"ariaHidden":90},[89],[59,3439,3441,3444],{"className":3440},[94],[59,3442],{"className":3443,"style":310},[98],[59,3445,3432],{"className":3446},[103,104]," 变换到 ",[59,3449,3451,3471],{"className":3450,"translate":63},[62],[59,3452,3454],{"className":3453},[67],[69,3455,3456],{"xmlns":71},[73,3457,3458,3468],{},[76,3459,3460],{},[132,3461,3462,3464],{},[79,3463,3432],{},[137,3465,3467],{"mathvariant":1091,"lspace":3466,"rspace":3466},"0em","′",[83,3469,3470],{"encoding":85},"x'",[59,3472,3474],{"className":3473,"ariaHidden":90},[89],[59,3475,3477,3481],{"className":3476},[94],[59,3478],{"className":3479,"style":3480},[98],"height:0.7519em;",[59,3482,3484,3487],{"className":3483},[103],[59,3485,3432],{"className":3486},[103,104],[59,3488,3490],{"className":3489},[162],[59,3491,3493],{"className":3492},[166],[59,3494,3496],{"className":3495},[170],[59,3497,3499],{"className":3498,"style":3480},[174],[59,3500,3501,3504],{"style":177},[59,3502],{"className":3503,"style":182},[181],[59,3505,3507],{"className":3506},[186,187,188,189],[59,3508,3510],{"className":3509},[103,189],[59,3511,3467],{"className":3512},[103,189],[59,3514,3516,3537],{"className":3515,"translate":63},[62],[59,3517,3519],{"className":3518},[67],[69,3520,3521],{"xmlns":71},[73,3522,3523,3535],{},[76,3524,3525,3527,3529,3531,3533],{},[137,3526,368],{"stretchy":367},[79,3528,14],{},[137,3530,1345],{"separator":90},[79,3532,1348],{},[137,3534,373],{"stretchy":367},[83,3536,1353],{"encoding":85},[59,3538,3540],{"className":3539,"ariaHidden":90},[89],[59,3541,3543,3546,3549,3552,3555,3558,3561],{"className":3542},[94],[59,3544],{"className":3545,"style":386},[98],[59,3547,368],{"className":3548},[393],[59,3550,14],{"className":3551},[103,104],[59,3553,1345],{"className":3554},[1372],[59,3556],{"className":3557,"style":983},[870],[59,3559,1348],{"className":3560,"style":314},[103,104],[59,3562,373],{"className":3563},[400]," 型张量的分量变换法则为：",[59,3566,3568],{"className":3567,"translate":63},[767],[59,3569,3571,3814],{"className":3570,"translate":63},[62],[59,3572,3574],{"className":3573},[67],[69,3575,3576],{"xmlns":71,"display":776},[73,3577,3578,3811],{},[76,3579,3580,3624,3626,3662,3664,3698,3700,3735,3737,3771,3773],{},[3581,3582,3583,3585,3604],"msubsup",{},[79,3584,1397],{},[76,3586,3587,3590,3596,3598],{},[1991,3588,3589],{},"  ",[2924,3591,3592,3594],{},[79,3593,2936],{},[473,3595,637],{},[137,3597,1513],{},[2924,3599,3600,3602],{},[79,3601,2936],{},[79,3603,1348],{},[76,3605,3606,3608],{},[137,3607,3467],{"mathvariant":1091},[76,3609,3610,3616,3618],{},[2924,3611,3612,3614],{},[79,3613,2933],{},[473,3615,637],{},[137,3617,1513],{},[2924,3619,3620,3622],{},[79,3621,2933],{},[79,3623,14],{},[137,3625,791],{},[1174,3627,3628,3647],{},[76,3629,3630,3633],{},[79,3631,3632],{"mathvariant":1091},"∂",[132,3634,3635,3637],{},[79,3636,3432],{},[76,3638,3639,3641],{},[137,3640,3467],{"mathvariant":1091},[2924,3642,3643,3645],{},[79,3644,2933],{},[473,3646,637],{},[76,3648,3649,3651],{},[79,3650,3632],{"mathvariant":1091},[132,3652,3653,3655],{},[79,3654,3432],{},[2924,3656,3657,3660],{},[79,3658,3659],{},"k",[473,3661,637],{},[137,3663,1513],{},[1174,3665,3666,3684],{},[76,3667,3668,3670],{},[79,3669,3632],{"mathvariant":1091},[132,3671,3672,3674],{},[79,3673,3432],{},[76,3675,3676,3678],{},[137,3677,3467],{"mathvariant":1091},[2924,3679,3680,3682],{},[79,3681,2933],{},[79,3683,14],{},[76,3685,3686,3688],{},[79,3687,3632],{"mathvariant":1091},[132,3689,3690,3692],{},[79,3691,3432],{},[2924,3693,3694,3696],{},[79,3695,3659],{},[79,3697,14],{},[137,3699,2316],{},[1174,3701,3702,3717],{},[76,3703,3704,3706],{},[79,3705,3632],{"mathvariant":1091},[132,3707,3708,3710],{},[79,3709,3432],{},[2924,3711,3712,3715],{},[79,3713,3714],{},"l",[473,3716,637],{},[76,3718,3719,3721],{},[79,3720,3632],{"mathvariant":1091},[132,3722,3723,3725],{},[79,3724,3432],{},[76,3726,3727,3729],{},[137,3728,3467],{"mathvariant":1091},[2924,3730,3731,3733],{},[79,3732,2936],{},[473,3734,637],{},[137,3736,1513],{},[1174,3738,3739,3753],{},[76,3740,3741,3743],{},[79,3742,3632],{"mathvariant":1091},[132,3744,3745,3747],{},[79,3746,3432],{},[2924,3748,3749,3751],{},[79,3750,3714],{},[79,3752,1348],{},[76,3754,3755,3757],{},[79,3756,3632],{"mathvariant":1091},[132,3758,3759,3761],{},[79,3760,3432],{},[76,3762,3763,3765],{},[137,3764,3467],{"mathvariant":1091},[2924,3766,3767,3769],{},[79,3768,2936],{},[79,3770,1348],{},[137,3772,2316],{},[3581,3774,3775,3777,3795],{},[79,3776,1397],{},[76,3778,3779,3781,3787,3789],{},[1991,3780,3589],{},[2924,3782,3783,3785],{},[79,3784,3714],{},[473,3786,637],{},[137,3788,1513],{},[2924,3790,3791,3793],{},[79,3792,3714],{},[79,3794,1348],{},[76,3796,3797,3803,3805],{},[2924,3798,3799,3801],{},[79,3800,3659],{},[473,3802,637],{},[137,3804,1513],{},[2924,3806,3807,3809],{},[79,3808,3659],{},[79,3810,14],{},[83,3812,3813],{"encoding":85},"T'^{i_1 \\cdots i_p}_{\\ \\ j_1 \\cdots j_q} = \\frac{\\partial x'^{i_1}}{\\partial x^{k_1}} \\cdots \\frac{\\partial x'^{i_p}}{\\partial x^{k_p}} \\cdot \\frac{\\partial x^{l_1}}{\\partial x'^{j_1}} \\cdots \\frac{\\partial x^{l_q}}{\\partial x'^{j_q}} \\cdot T^{k_1 \\cdots k_p}_{\\ \\ l_1 \\cdots 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x}",[59,5254,5256],{"className":5255,"ariaHidden":90},[89],[59,5257,5259,5263],{"className":5258},[94],[59,5260],{"className":5261,"style":5262},[98],"height:1.3185em;vertical-align:-0.345em;",[59,5264,5266,5269,5366],{"className":5265},[103],[59,5267],{"className":5268},[393,1222],[59,5270,5272],{"className":5271},[1174],[59,5273,5275,5358],{"className":5274},[166,527],[59,5276,5278,5355],{"className":5277},[170],[59,5279,5282,5299,5307],{"className":5280,"style":5281},[174],"height:0.9735em;",[59,5283,5284,5287],{"style":1238},[59,5285],{"className":5286,"style":541},[181],[59,5288,5290],{"className":5289},[186,187,188,189],[59,5291,5293,5296],{"className":5292},[103,189],[59,5294,3632],{"className":5295,"style":4119},[103,189],[59,5297,3432],{"className":5298},[103,104,189],[59,5300,5301,5304],{"style":1253},[59,5302],{"className":5303,"style":541},[181],[59,5305],{"className":5306,"style":1261},[1260],[59,5308,5309,5312],{"style":1264},[59,5310],{"className":5311,"style":541},[181],[59,5313,5315],{"className":5314},[186,187,188,189],[59,5316,5318,5321],{"className":5317},[103,189],[59,5319,3632],{"className":5320,"style":4119},[103,189],[59,5322,5324,5327],{"className":5323},[103,189],[59,5325,3432],{"className":5326},[103,104,189],[59,5328,5330],{"className":5329},[162],[59,5331,5333],{"className":5332},[166],[59,5334,5336],{"className":5335},[170],[59,5337,5340],{"className":5338,"style":5339},[174],"height:0.8278em;",[59,5341,5343,5346],{"style":5342},"top:-2.931em;margin-right:0.0714em;",[59,5344],{"className":5345,"style":3893},[181],[59,5347,5349],{"className":5348},[186,3897,669,189],[59,5350,5352],{"className":5351},[103,189],[59,5353,3467],{"className":5354},[103,189],[59,5356,564],{"className":5357},[563],[59,5359,5361],{"className":5360},[170],[59,5362,5364],{"className":5363,"style":1286},[174],[59,5365],{},[59,5367],{"className":5368},[400,1222]," 的雅可比因子；",[409,5371,5372,5373,5522],{},"每个下标（对应对偶向量槽位的「协变」分量）乘以一个形如 ",[59,5374,5376,5406],{"className":5375,"translate":63},[62],[59,5377,5379],{"className":5378},[67],[69,5380,5381],{"xmlns":71},[73,5382,5383,5403],{},[76,5384,5385],{},[1174,5386,5387,5393],{},[76,5388,5389,5391],{},[79,5390,3632],{"mathvariant":1091},[79,5392,3432],{},[76,5394,5395,5397],{},[79,5396,3632],{"mathvariant":1091},[132,5398,5399,5401],{},[79,5400,3432],{},[137,5402,3467],{"mathvariant":1091,"lspace":3466,"rspace":3466},[83,5404,5405],{"encoding":85},"\\frac{\\partial x}{\\partial x'}",[59,5407,5409],{"className":5408,"ariaHidden":90},[89],[59,5410,5412,5416],{"className":5411},[94],[59,5413],{"className":5414,"style":5415},[98],"height:1.2251em;vertical-align:-0.345em;",[59,5417,5419,5422,5519],{"className":5418},[103],[59,5420],{"className":5421},[393,1222],[59,5423,5425],{"className":5424},[1174],[59,5426,5428,5511],{"className":5427},[166,527],[59,5429,5431,5508],{"className":5430},[170],[59,5432,5435,5483,5491],{"className":5433,"style":5434},[174],"height:0.8801em;",[59,5436,5437,5440],{"style":1238},[59,5438],{"className":5439,"style":541},[181],[59,5441,5443],{"className":5442},[186,187,188,189],[59,5444,5446,5449],{"className":5445},[103,189],[59,5447,3632],{"className":5448,"style":4119},[103,189],[59,5450,5452,5455],{"className":5451},[103,189],[59,5453,3432],{"className":5454},[103,104,189],[59,5456,5458],{"className":5457},[162],[59,5459,5461],{"className":5460},[166],[59,5462,5464],{"className":5463},[170],[59,5465,5468],{"className":5466,"style":5467},[174],"height:0.6828em;",[59,5469,5471,5474],{"style":5470},"top:-2.786em;margin-right:0.0714em;",[59,5472],{"className":5473,"style":3893},[181],[59,5475,5477],{"className":5476},[186,3897,669,189],[59,5478,5480],{"className":5479},[103,189],[59,5481,3467],{"className":5482},[103,189],[59,5484,5485,5488],{"style":1253},[59,5486],{"className":5487,"style":541},[181],[59,5489],{"className":5490,"style":1261},[1260],[59,5492,5493,5496],{"style":1264},[59,5494],{"className":5495,"style":541},[181],[59,5497,5499],{"className":5498},[186,187,188,189],[59,5500,5502,5505],{"className":5501},[103,189],[59,5503,3632],{"className":5504,"style":4119},[103,189],[59,5506,3432],{"className":5507},[103,104,189],[59,5509,564],{"className":5510},[563],[59,5512,5514],{"className":5513},[170],[59,5515,5517],{"className":5516,"style":1286},[174],[59,5518],{},[59,5520],{"className":5521},[400,1222]," 的雅可比因子（正是前者的倒数）。",[14,5524,5525],{},"这就是第 2 章那个直观例子「厘米与米」的严格数学版：上下指标的雅可比因子互为倒数，配对缩并时精确抵消，从而保证张量整体（作为几何对象）不变，只有相对于特定坐标系的分量在变。",[30,5527,5528],{"id":5528},"常见张量",[14,5530,5531],{},"下面列举各个领域中遇到的张量示例。",[5533,5534,5535,5613],"table",{},[5536,5537,5538],"thead",{},[5539,5540,5541,5545,5576,5607,5610],"tr",{},[5542,5543,5544],"th",{},"张量类型",[5542,5546,5547,5575],{},[59,5548,5550,5563],{"className":5549,"translate":63},[62],[59,5551,5553],{"className":5552},[67],[69,5554,5555],{"xmlns":71},[73,5556,5557,5561],{},[76,5558,5559],{},[79,5560,14],{},[83,5562,14],{"encoding":85},[59,5564,5566],{"className":5565,"ariaHidden":90},[89],[59,5567,5569,5572],{"className":5568},[94],[59,5570],{"className":5571,"style":1439},[98],[59,5573,14],{"className":5574},[103,104],"（对偶向量槽位）",[5542,5577,5578,5606],{},[59,5579,5581,5594],{"className":5580,"translate":63},[62],[59,5582,5584],{"className":5583},[67],[69,5585,5586],{"xmlns":71},[73,5587,5588,5592],{},[76,5589,5590],{},[79,5591,1348],{},[83,5593,1348],{"encoding":85},[59,5595,5597],{"className":5596,"ariaHidden":90},[89],[59,5598,5600,5603],{"className":5599},[94],[59,5601],{"className":5602,"style":1439},[98],[59,5604,1348],{"className":5605,"style":314},[103,104],"（向量槽位）",[5542,5608,5609],{},"总阶数",[5542,5611,5612],{},"典型例子",[5614,5615,5616,5631,5645,5659,5673,5904],"tbody",{},[5539,5617,5618,5622,5624,5626,5628],{},[5619,5620,5621],"td",{},"标量",[5619,5623,471],{},[5619,5625,471],{},[5619,5627,471],{},[5619,5629,5630],{},"温度、质量",[5539,5632,5633,5636,5638,5640,5642],{},[5619,5634,5635],{},"向量",[5619,5637,471],{},[5619,5639,637],{},[5619,5641,637],{},[5619,5643,5644],{},"速度、力",[5539,5646,5647,5650,5652,5654,5656],{},[5619,5648,5649],{},"对偶向量（1-形式）",[5619,5651,637],{},[5619,5653,471],{},[5619,5655,637],{},[5619,5657,5658],{},"梯度、行向量",[5539,5660,5661,5664,5666,5668,5670],{},[5619,5662,5663],{},"线性变换\u002F惯性张量",[5619,5665,637],{},[5619,5667,637],{},[5619,5669,475],{},[5619,5671,5672],{},"转动惯量、应力张量",[5539,5674,5675,5677,5679,5681,5683],{},[5619,5676,2904],{},[5619,5678,471],{},[5619,5680,475],{},[5619,5682,475],{},[5619,5684,5685,5686],{},"长度定义 ",[59,5687,5689,5739],{"className":5688,"translate":63},[62],[59,5690,5692],{"className":5691},[67],[69,5693,5694],{"xmlns":71},[73,5695,5696,5736],{},[76,5697,5698,5701,5708,5710,5720,5722,5728,5730],{},[79,5699,5700],{},"d",[132,5702,5703,5706],{},[79,5704,5705],{},"s",[473,5707,475],{},[137,5709,791],{},[2924,5711,5712,5714],{},[79,5713,2928],{},[76,5715,5716,5718],{},[79,5717,2933],{},[79,5719,2936],{},[79,5721,5700],{},[132,5723,5724,5726],{},[79,5725,3432],{},[79,5727,2933],{},[79,5729,5700],{},[132,5731,5732,5734],{},[79,5733,3432],{},[79,5735,2936],{},[83,5737,5738],{"encoding":85},"ds^2 = g_{ij}dx^i dx^j",[59,5740,5742,5790],{"className":5741,"ariaHidden":90},[89],[59,5743,5745,5749,5752,5781,5784,5787],{"className":5744},[94],[59,5746],{"className":5747,"style":5748},[98],"height:0.8141em;",[59,5750,5700],{"className":5751},[103,104],[59,5753,5755,5758],{"className":5754},[103],[59,5756,5705],{"className":5757},[103,104],[59,5759,5761],{"className":5760},[162],[59,5762,5764],{"className":5763},[166],[59,5765,5767],{"className":5766},[170],[59,5768,5770],{"className":5769,"style":5748},[174],[59,5771,5772,5775],{"style":177},[59,5773],{"className":5774,"style":182},[181],[59,5776,5778],{"className":5777},[186,187,188,189],[59,5779,475],{"className":5780},[103,189],[59,5782],{"className":5783,"style":871},[870],[59,5785,791],{"className":5786},[875],[59,5788],{"className":5789,"style":871},[870],[59,5791,5793,5797,5840,5843,5872,5875],{"className":5792},[94],[59,5794],{"className":5795,"style":5796},[98],"height:1.1108em;vertical-align:-0.2861em;",[59,5798,5800,5803],{"className":5799},[103],[59,5801,2928],{"className":5802,"style":314},[103,104],[59,5804,5806],{"className":5805},[162],[59,5807,5809,5832],{"className":5808},[166,527],[59,5810,5812,5829],{"className":5811},[170],[59,5813,5815],{"className":5814,"style":2968},[174],[59,5816,5817,5820],{"style":2971},[59,5818],{"className":5819,"style":182},[181],[59,5821,5823],{"className":5822},[186,187,188,189],[59,5824,5826],{"className":5825},[103,189],[59,5827,2985],{"className":5828,"style":2984},[103,104,189],[59,5830,564],{"className":5831},[563],[59,5833,5835],{"className":5834},[170],[59,5836,5838],{"className":5837,"style":2995},[174],[59,5839],{},[59,5841,5700],{"className":5842},[103,104],[59,5844,5846,5849],{"className":5845},[103],[59,5847,3432],{"className":5848},[103,104],[59,5850,5852],{"className":5851},[162],[59,5853,5855],{"className":5854},[166],[59,5856,5858],{"className":5857},[170],[59,5859,5861],{"className":5860,"style":4231},[174],[59,5862,5863,5866],{"style":177},[59,5864],{"className":5865,"style":182},[181],[59,5867,5869],{"className":5868},[186,187,188,189],[59,5870,2933],{"className":5871},[103,104,189],[59,5873,5700],{"className":5874},[103,104],[59,5876,5878,5881],{"className":5877},[103],[59,5879,3432],{"className":5880},[103,104],[59,5882,5884],{"className":5883},[162],[59,5885,5887],{"className":5886},[166],[59,5888,5890],{"className":5889},[170],[59,5891,5893],{"className":5892,"style":4231},[174],[59,5894,5895,5898],{"style":177},[59,5896],{"className":5897,"style":182},[181],[59,5899,5901],{"className":5900},[186,187,188,189],[59,5902,2936],{"className":5903,"style":2984},[103,104,189],[5539,5905,5906,5909,5911,5913,5915],{},[5619,5907,5908],{},"黎曼曲率张量",[5619,5910,637],{},[5619,5912,484],{},[5619,5914,644],{},[5619,5916,5917],{},"广义相对论中的时空弯曲",[30,5919,5920],{"id":5920},"两种语境",[14,5922,5923],{},"最后需要澄清一点：「张量」一词在深度学习和在物理\u002F微分几何中，指的是两个不同抽象层级上的概念，这是困惑的重要来源。",[406,5925,5926,5933],{},[409,5927,5928,5929,5932],{},"深度学习：",[40,5930,5931],{},"tensor"," 只是一个多维数组；唯一关心的是并行计算的效率。这里没有「坐标变换不变性」，也没有协变与逆变之分。把张量理解成「广义数组」就完全够用，不必给这个词背上哲学重负。",[409,5934,5935],{},"物理与微分几何：张量首先是一个不依赖任何坐标选择的几何或物理对象；数组只是它相对于某个坐标基的「临时快照」。协变、逆变、度量、缩并——这些概念存在的目的只有一个：保证从这些临时快照算出的物理量（长度、能量、曲率）是客观的、绝对的，不随观察者改变。",[14,5937,5938],{},"两种语境没有对错之分；它们服务的目的大相径庭。真正的困惑往往来自用深度学习的直觉去理解广义相对论的张量，或者反过来。一旦意识到这是两回事，很多「教科书讲不清楚」式的挫败感就会自行消解。",[30,5940,5941],{"id":5941},"结语",[14,5943,5944],{},"让我们回到最初的问题：张量到底是什么？",[14,5946,5947],{},"它是一台带有若干「槽位」的机器：有的槽位等待「箭头」（向量），有的槽位等待「标尺」（对偶向量）。当所有槽位都被填满时，它输出的数字永远相同——无论你从哪个坐标系观察，无论你用什么单位度量。而度量张量，则是连接向量世界与对偶向量世界的桥梁，使「长度」这样一个看似无辜的概念得以被严格定义。",[14,5949,5950],{},"下次你写下下标或上标时，不妨停下来想一想：这个位置是在等一支箭头，还是在等一把标尺？",{"title":5952,"searchDepth":5953,"depth":5953,"links":5954},"",2,[5955,5956,5957,5958,5959,5960,5961,5962,5963],{"id":32,"depth":5953,"text":32},{"id":109,"depth":5953,"text":109},{"id":1319,"depth":5953,"text":1319},{"id":2065,"depth":5953,"text":2065},{"id":2904,"depth":5953,"text":2904},{"id":3414,"depth":5953,"text":3414},{"id":5528,"depth":5953,"text":5528},{"id":5920,"depth":5953,"text":5920},{"id":5941,"depth":5953,"text":5941},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F2026-05-30-about-the-tensor",{"title":6,"description":5952},"blog\u002F2026\u002F2026-05-30-about-the-tensor","向量是箭头，对偶向量是标尺——它们的配对产生不变的标量。张量是带多个可填充槽位的线性机器，化解了「向量进向量出」与「输出标量」之间的悖论。度量张量连接二者，使长度与能量保持坐标不变。",[5974,5975],"mathematics","machine-learning","2026-05-30T00:00:00+08:00","tt888YdfwWUrGoFGv7BNbjumPfXXrLFHBTFRPg7fvjA",{"id":5979,"title":5980,"body":5981,"description":5952,"draft":5964,"enableComment":5965,"extension":5966,"image":5952,"important":5964,"location":5967,"meta":21001,"navigation":5965,"ogImage":5967,"path":21002,"seo":21003,"stem":21004,"summary":21005,"tags":21006,"time":21007,"__hash__":21008},"blog\u002Fblog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory.md","魏尔斯特拉斯理论下的 UAT 证明",{"type":8,"value":5982,"toc":20992},[5983,5986,5990,5993,6087,6172,6175,6232,6281,6477,6480,6483,6486,7013,7016,7019,7235,7267,7270,7373,7376,7379,7382,7821,7823,7925,7928,8129,8132,8449,8452,8455,8458,8615,8618,8939,8942,9250,9253,9418,9421,9468,9471,9668,9974,9976,10113,10145,10202,10365,10368,10581,10583,10922,10925,10928,11095,11183,11186,11371,11374,11767,11770,11957,11960,12113,12116,12305,12442,12677,12780,12931,13108,13278,13281,13722,13724,14026,14057,14284,14286,14463,14495,14498,14767,14769,14879,14882,14885,14889,14892,15046,15381,15384,15569,15601,15604,15651,15856,16122,16184,16289,16292,16615,16663,16857,16860,17227,17230,17412,17669,17731,17736,17739,18115,18385,18388,18640,18643,19296,19299,19302,19551,19554,19896,19899,19902,19906,19909,19912,20456,20459,20762,20765,20768,20989],[30,5984,5985],{"id":5985},"魏尔斯特拉斯定理",[5987,5988,5989],"h3",{"id":5989},"直观解释",[14,5991,5992],{},"假设",[59,5994,5996],{"className":5995,"translate":63},[767],[59,5997,5999,6034],{"className":5998,"translate":63},[62],[59,6000,6002],{"className":6001},[67],[69,6003,6004],{"xmlns":71,"display":776},[73,6005,6006,6031],{},[76,6007,6008,6011,6014,6017,6019,6021,6023,6025,6027,6029],{},[79,6009,6010],{},"f",[137,6012,6013],{},"∈",[79,6015,6016],{},"C",[137,6018,368],{"stretchy":367},[137,6020,455],{"stretchy":367},[473,6022,471],{},[137,6024,1345],{"separator":90},[473,6026,637],{},[137,6028,487],{"stretchy":367},[137,6030,373],{"stretchy":367},[83,6032,6033],{"encoding":85},"f\\in C([0,1])",[59,6035,6037,6057],{"className":6036,"ariaHidden":90},[89],[59,6038,6040,6044,6048,6051,6054],{"className":6039},[94],[59,6041],{"className":6042,"style":6043},[98],"height:0.8889em;vertical-align:-0.1944em;",[59,6045,6010],{"className":6046,"style":6047},[103,104],"margin-right:0.1076em;",[59,6049],{"className":6050,"style":871},[870],[59,6052,6013],{"className":6053},[875],[59,6055],{"className":6056,"style":871},[870],[59,6058,6060,6063,6067,6071,6074,6077,6080,6083],{"className":6059},[94],[59,6061],{"className":6062,"style":386},[98],[59,6064,6016],{"className":6065,"style":6066},[103,104],"margin-right:0.0715em;",[59,6068,6070],{"className":6069},[393],"([",[59,6072,471],{"className":6073},[103],[59,6075,1345],{"className":6076},[1372],[59,6078],{"className":6079,"style":983},[870],[59,6081,637],{"className":6082},[103],[59,6084,6086],{"className":6085},[400],"])",[14,6088,6089,6090,6118,6119,6171],{},"即 ",[59,6091,6093,6106],{"className":6092,"translate":63},[62],[59,6094,6096],{"className":6095},[67],[69,6097,6098],{"xmlns":71},[73,6099,6100,6104],{},[76,6101,6102],{},[79,6103,6010],{},[83,6105,6010],{"encoding":85},[59,6107,6109],{"className":6108,"ariaHidden":90},[89],[59,6110,6112,6115],{"className":6111},[94],[59,6113],{"className":6114,"style":6043},[98],[59,6116,6010],{"className":6117,"style":6047},[103,104]," 在 ",[59,6120,6122,6144],{"className":6121,"translate":63},[62],[59,6123,6125],{"className":6124},[67],[69,6126,6127],{"xmlns":71},[73,6128,6129,6141],{},[76,6130,6131,6133,6135,6137,6139],{},[137,6132,455],{"stretchy":367},[473,6134,471],{},[137,6136,1345],{"separator":90},[473,6138,637],{},[137,6140,487],{"stretchy":367},[83,6142,6143],{"encoding":85},"[0,1]",[59,6145,6147],{"className":6146,"ariaHidden":90},[89],[59,6148,6150,6153,6156,6159,6162,6165,6168],{"className":6149},[94],[59,6151],{"className":6152,"style":386},[98],[59,6154,455],{"className":6155},[393],[59,6157,471],{"className":6158},[103],[59,6160,1345],{"className":6161},[1372],[59,6163],{"className":6164,"style":983},[870],[59,6166,637],{"className":6167},[103],[59,6169,487],{"className":6170},[400]," 上连续。",[14,6173,6174],{},"那么对任意",[59,6176,6178],{"className":6177,"translate":63},[767],[59,6179,6181,6201],{"className":6180,"translate":63},[62],[59,6182,6184],{"className":6183},[67],[69,6185,6186],{"xmlns":71,"display":776},[73,6187,6188,6198],{},[76,6189,6190,6193,6196],{},[79,6191,6192],{},"ε",[137,6194,6195],{},">",[473,6197,471],{},[83,6199,6200],{"encoding":85},"\\varepsilon>0",[59,6202,6204,6223],{"className":6203,"ariaHidden":90},[89],[59,6205,6207,6211,6214,6217,6220],{"className":6206},[94],[59,6208],{"className":6209,"style":6210},[98],"height:0.5782em;vertical-align:-0.0391em;",[59,6212,6192],{"className":6213},[103,104],[59,6215],{"className":6216,"style":871},[870],[59,6218,6195],{"className":6219},[875],[59,6221],{"className":6222,"style":871},[870],[59,6224,6226,6229],{"className":6225},[94],[59,6227],{"className":6228,"style":1065},[98],[59,6230,471],{"className":6231},[103],[14,6233,6234,6235,6280],{},"存在多项式 ",[59,6236,6238,6259],{"className":6237,"translate":63},[62],[59,6239,6241],{"className":6240},[67],[69,6242,6243],{"xmlns":71},[73,6244,6245,6256],{},[76,6246,6247,6250,6252,6254],{},[79,6248,6249],{},"P",[137,6251,368],{"stretchy":367},[79,6253,3432],{},[137,6255,373],{"stretchy":367},[83,6257,6258],{"encoding":85},"P(x)",[59,6260,6262],{"className":6261,"ariaHidden":90},[89],[59,6263,6265,6268,6271,6274,6277],{"className":6264},[94],[59,6266],{"className":6267,"style":386},[98],[59,6269,6249],{"className":6270,"style":1412},[103,104],[59,6272,368],{"className":6273},[393],[59,6275,3432],{"className":6276},[103,104],[59,6278,373],{"className":6279},[400]," 使得",[59,6282,6284],{"className":6283,"translate":63},[767],[59,6285,6287,6353],{"className":6286,"translate":63},[62],[59,6288,6290],{"className":6289},[67],[69,6291,6292],{"xmlns":71,"display":776},[73,6293,6294,6350],{},[76,6295,6296,6299,6301,6303,6305,6307,6310,6312,6314,6316,6318,6320,6323,6325,6327,6330,6333,6335,6337,6339,6341,6343,6345,6347],{},[79,6297,6298],{"mathvariant":1091},"∣",[79,6300,6010],{},[137,6302,368],{"stretchy":367},[79,6304,3432],{},[137,6306,373],{"stretchy":367},[137,6308,6309],{},"−",[79,6311,6249],{},[137,6313,368],{"stretchy":367},[79,6315,3432],{},[137,6317,373],{"stretchy":367},[79,6319,6298],{"mathvariant":1091},[137,6321,6322],{},"\u003C",[79,6324,6192],{},[137,6326,1345],{"separator":90},[870,6328],{"width":6329},"2em",[79,6331,6332],{"mathvariant":1091},"∀",[79,6334,3432],{},[137,6336,6013],{},[137,6338,455],{"stretchy":367},[473,6340,471],{},[137,6342,1345],{"separator":90},[473,6344,637],{},[137,6346,487],{"stretchy":367},[79,6348,6349],{"mathvariant":1091},".",[83,6351,6352],{"encoding":85},"|f(x)-P(x)|\u003C\\varepsilon,\n\\qquad\n\\forall x\\in [0,1].",[59,6354,6356,6386,6416,6450],{"className":6355,"ariaHidden":90},[89],[59,6357,6359,6362,6365,6368,6371,6374,6377,6380,6383],{"className":6358},[94],[59,6360],{"className":6361,"style":386},[98],[59,6363,6298],{"className":6364},[103],[59,6366,6010],{"className":6367,"style":6047},[103,104],[59,6369,368],{"className":6370},[393],[59,6372,3432],{"className":6373},[103,104],[59,6375,373],{"className":6376},[400],[59,6378],{"className":6379,"style":105},[870],[59,6381,6309],{"className":6382},[193],[59,6384],{"className":6385,"style":105},[870],[59,6387,6389,6392,6395,6398,6401,6404,6407,6410,6413],{"className":6388},[94],[59,6390],{"className":6391,"style":386},[98],[59,6393,6249],{"className":6394,"style":1412},[103,104],[59,6396,368],{"className":6397},[393],[59,6399,3432],{"className":6400},[103,104],[59,6402,373],{"className":6403},[400],[59,6405,6298],{"className":6406},[103],[59,6408],{"className":6409,"style":871},[870],[59,6411,6322],{"className":6412},[875],[59,6414],{"className":6415,"style":871},[870],[59,6417,6419,6422,6425,6428,6432,6435,6438,6441,6444,6447],{"className":6418},[94],[59,6420],{"className":6421,"style":6043},[98],[59,6423,6192],{"className":6424},[103,104],[59,6426,1345],{"className":6427},[1372],[59,6429],{"className":6430,"style":6431},[870],"margin-right:2em;",[59,6433],{"className":6434,"style":983},[870],[59,6436,6332],{"className":6437},[103],[59,6439,3432],{"className":6440},[103,104],[59,6442],{"className":6443,"style":871},[870],[59,6445,6013],{"className":6446},[875],[59,6448],{"className":6449,"style":871},[870],[59,6451,6453,6456,6459,6462,6465,6468,6471,6474],{"className":6452},[94],[59,6454],{"className":6455,"style":386},[98],[59,6457,455],{"className":6458},[393],[59,6460,471],{"className":6461},[103],[59,6463,1345],{"className":6464},[1372],[59,6466],{"className":6467,"style":983},[870],[59,6469,637],{"className":6470},[103],[59,6472,487],{"className":6473},[400],[59,6475,6349],{"className":6476},[103],[14,6478,6479],{},"接下来，我们尝试证明这个定理。",[14,6481,6482],{},"第 1 步：构造伯恩斯坦多项式",[14,6484,6485],{},"定义",[59,6487,6489],{"className":6488,"translate":63},[767],[59,6490,6492,6599],{"className":6491,"translate":63},[62],[59,6493,6495],{"className":6494},[67],[69,6496,6497],{"xmlns":71,"display":776},[73,6498,6499,6596],{},[76,6500,6501,6508,6510,6512,6514,6516,6518,6520,6522,6538,6540,6543,6555,6568,6574,6576,6578,6580,6582,6594],{},[2924,6502,6503,6506],{},[79,6504,6505],{},"B",[79,6507,2119],{},[137,6509,368],{"stretchy":367},[79,6511,6010],{},[137,6513,373],{"stretchy":367},[137,6515,368],{"stretchy":367},[79,6517,3432],{},[137,6519,373],{"stretchy":367},[137,6521,791],{},[6523,6524,6525,6528,6536],"munderover",{},[137,6526,6527],{},"∑",[76,6529,6530,6532,6534],{},[79,6531,3659],{},[137,6533,791],{},[473,6535,471],{},[79,6537,2119],{},[79,6539,6010],{},[1991,6541,6542],{}," ⁣",[76,6544,6545,6547,6553],{},[137,6546,368],{"fence":90},[1174,6548,6549,6551],{},[79,6550,3659],{},[79,6552,2119],{},[137,6554,373],{"fence":90},[76,6556,6557,6559,6566],{},[137,6558,368],{"fence":90},[1174,6560,6562,6564],{"linethickness":6561},"0px",[79,6563,2119],{},[79,6565,3659],{},[137,6567,373],{"fence":90},[132,6569,6570,6572],{},[79,6571,3432],{},[79,6573,3659],{},[137,6575,368],{"stretchy":367},[473,6577,637],{},[137,6579,6309],{},[79,6581,3432],{},[132,6583,6584,6586],{},[137,6585,373],{"stretchy":367},[76,6587,6588,6590,6592],{},[79,6589,2119],{},[137,6591,6309],{},[79,6593,3659],{},[79,6595,6349],{"mathvariant":1091},[83,6597,6598],{"encoding":85},"B_n(f)(x)\n=\n\n\\sum_{k=0}^{n}\nf\\!\\left(\\frac{k}{n}\\right)\n\\binom{n}{k}\nx^k(1-x)^{n-k}.",[59,6600,6602,6678,6962],{"className":6601,"ariaHidden":90},[89],[59,6603,6605,6608,6651,6654,6657,6660,6663,6666,6669,6672,6675],{"className":6604},[94],[59,6606],{"className":6607,"style":386},[98],[59,6609,6611,6615],{"className":6610},[103],[59,6612,6505],{"className":6613,"style":6614},[103,104],"margin-right:0.0502em;",[59,6616,6618],{"className":6617},[162],[59,6619,6621,6643],{"className":6620},[166,527],[59,6622,6624,6640],{"className":6623},[170],[59,6625,6628],{"className":6626,"style":6627},[174],"height:0.1514em;",[59,6629,6631,6634],{"style":6630},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[59,6632],{"className":6633,"style":182},[181],[59,6635,6637],{"className":6636},[186,187,188,189],[59,6638,2119],{"className":6639},[103,104,189],[59,6641,564],{"className":6642},[563],[59,6644,6646],{"className":6645},[170],[59,6647,6649],{"className":6648,"style":3092},[174],[59,6650],{},[59,6652,368],{"className":6653},[393],[59,6655,6010],{"className":6656,"style":6047},[103,104],[59,6658,373],{"className":6659},[400],[59,6661,368],{"className":6662},[393],[59,6664,3432],{"className":6665},[103,104],[59,6667,373],{"className":6668},[400],[59,6670],{"className":6671,"style":871},[870],[59,6673,791],{"className":6674},[875],[59,6676],{"className":6677,"style":871},[870],[59,6679,6681,6685,6762,6765,6768,6772,6775,6853,6856,6917,6947,6950,6953,6956,6959],{"className":6680},[94],[59,6682],{"className":6683,"style":6684},[98],"height:2.9535em;vertical-align:-1.3021em;",[59,6686,6690],{"className":6687},[6688,6689],"mop","op-limits",[59,6691,6693,6753],{"className":6692},[166,527],[59,6694,6696,6750],{"className":6695},[170],[59,6697,6700,6722,6735],{"className":6698,"style":6699},[174],"height:1.6514em;",[59,6701,6703,6707],{"style":6702},"top:-1.8479em;margin-left:0em;",[59,6704],{"className":6705,"style":6706},[181],"height:3.05em;",[59,6708,6710],{"className":6709},[186,187,188,189],[59,6711,6713,6716,6719],{"className":6712},[103,189],[59,6714,3659],{"className":6715,"style":4157},[103,104,189],[59,6717,791],{"className":6718},[875,189],[59,6720,471],{"className":6721},[103,189],[59,6723,6725,6728],{"style":6724},"top:-3.05em;",[59,6726],{"className":6727,"style":6706},[181],[59,6729,6730],{},[59,6731,6527],{"className":6732},[6688,6733,6734],"op-symbol","large-op",[59,6736,6738,6741],{"style":6737},"top:-4.3em;margin-left:0em;",[59,6739],{"className":6740,"style":6706},[181],[59,6742,6744],{"className":6743},[186,187,188,189],[59,6745,6747],{"className":6746},[103,189],[59,6748,2119],{"className":6749},[103,104,189],[59,6751,564],{"className":6752},[563],[59,6754,6756],{"className":6755},[170],[59,6757,6760],{"className":6758,"style":6759},[174],"height:1.3021em;",[59,6761],{},[59,6763],{"className":6764,"style":983},[870],[59,6766,6010],{"className":6767,"style":6047},[103,104],[59,6769],{"className":6770,"style":6771},[870],"margin-right:-0.1667em;",[59,6773],{"className":6774,"style":983},[870],[59,6776,6778,6784,6847],{"className":6777},[504],[59,6779,6781],{"className":6780,"style":509},[393,508],[59,6782,368],{"className":6783},[513,188],[59,6785,6787,6790,6844],{"className":6786},[103],[59,6788],{"className":6789},[393,1222],[59,6791,6793],{"className":6792},[1174],[59,6794,6796,6836],{"className":6795},[166,527],[59,6797,6799,6833],{"className":6798},[170],[59,6800,6803,6814,6822],{"className":6801,"style":6802},[174],"he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],{"className":7129},[563],[59,7131,7133],{"className":7132},[170],[59,7134,7136],{"className":7135,"style":4298},[174],[59,7137],{},[59,7139,7141],{"className":7140,"style":509},[400,508],[59,7142,373],{"className":7143},[513,188],[59,7145,7147,7150],{"className":7146},[103],[59,7148,3432],{"className":7149},[103,104],[59,7151,7153],{"className":7152},[162],[59,7154,7156],{"className":7155},[166],[59,7157,7159],{"className":7158},[170],[59,7160,7162],{"className":7161,"style":6935},[174],[59,7163,7164,7167],{"style":1711},[59,7165],{"className":7166,"style":182},[181],[59,7168,7170],{"className":7169},[186,187,188,189],[59,7171,3659],{"className":7172,"style":4157},[103,104,189],[59,7174,368],{"className":7175},[393],[59,7177,637],{"className":7178},[103],[59,7180],{"className":7181,"style":105},[870],[59,7183,6309],{"className":7184},[193],[59,7186],{"className":7187,"style":105},[870],[59,7189,7191,7194,7197],{"className":7190},[94],[59,7192],{"className":7193,"style":6968},[98],[59,7195,3432],{"className":7196},[103,104],[59,7198,7200,7203],{"className":7199},[400],[59,7201,373],{"className":7202},[400],[59,7204,7206],{"className":7205},[162],[59,7207,7209],{"className":7208},[166],[59,7210,7212],{"className":7211},[170],[59,7213,7215],{"className":7214,"style":6935},[174],[59,7216,7217,7220],{"style":1711},[59,7218],{"className":7219,"style":182},[181],[59,7221,7223],{"className":7222},[186,187,188,189],[59,7224,7226,7229,7232],{"className":7225},[103,189],[59,7227,2119],{"className":7228},[103,104,189],[59,7230,6309],{"className":7231},[193,189],[59,7233,3659],{"className":7234,"style":4157},[103,104,189],[14,7236,7237,7238,7266],{},"本身是 ",[59,7239,7241,7254],{"className":7240,"translate":63},[62],[59,7242,7244],{"className":7243},[67],[69,7245,7246],{"xmlns":71},[73,7247,7248,7252],{},[76,7249,7250],{},[79,7251,3432],{},[83,7253,3432],{"encoding":85},[59,7255,7257],{"className":7256,"ariaHidden":90},[89],[59,7258,7260,7263],{"className":7259},[94],[59,7261],{"className":7262,"style":310},[98],[59,7264,3432],{"className":7265},[103,104]," 的多项式。",[14,7268,7269],{},"因此",[59,7271,7273],{"className":7272,"translate":63},[767],[59,7274,7276,7306],{"className":7275,"translate":63},[62],[59,7277,7279],{"className":7278},[67],[69,7280,7281],{"xmlns":71,"display":776},[73,7282,7283,7303],{},[76,7284,7285,7291,7293,7295,7297,7299,7301],{},[2924,7286,7287,7289],{},[79,7288,6505],{},[79,7290,2119],{},[137,7292,368],{"stretchy":367},[79,7294,6010],{},[137,7296,373],{"stretchy":367},[137,7298,368],{"stretchy":367},[79,7300,3432],{},[137,7302,373],{"stretchy":367},[83,7304,7305],{"encoding":85},"B_n(f)(x)",[59,7307,7309],{"className":7308,"ariaHidden":90},[89],[59,7310,7312,7315,7355,7358,7361,7364,7367,7370],{"className":7311},[94],[59,7313],{"className":7314,"style":386},[98],[59,7316,7318,7321],{"className":7317},[103],[59,7319,6505],{"className":7320,"style":6614},[103,104],[59,7322,7324],{"className":7323},[162],[59,7325,7327,7347],{"className":7326},[166,527],[59,7328,7330,7344],{"className":7329},[170],[59,7331,7333],{"className":7332,"style":6627},[174],[59,7334,7335,7338],{"style":6630},[59,7336],{"className":7337,"style":182},[181],[59,7339,7341],{"className":7340},[186,187,188,189],[59,7342,2119],{"className":7343},[103,104,189],[59,7345,564],{"className":7346},[563],[59,7348,7350],{"className":7349},[170],[59,7351,7353],{"className":7352,"style":3092},[174],[59,7354],{},[59,7356,368],{"className":7357},[393],[59,7359,6010],{"className":7360,"style":6047},[103,104],[59,7362,373],{"className":7363},[400],[59,7365,368],{"className":7366},[393],[59,7368,3432],{"className":7369},[103,104],[59,7371,373],{"className":7372},[400],[14,7374,7375],{},"也是多项式。",[14,7377,7378],{},"第 2 步：理解它在做什么。这个多项式有点抽象，但我们可以换一种方式解读它。",[14,7380,7381],{},"首先注意恒等式",[59,7383,7385],{"className":7384,"translate":63},[767],[59,7386,7388,7480],{"className":7387,"translate":63},[62],[59,7389,7391],{"className":7390},[67],[69,7392,7393],{"xmlns":71,"display":776},[73,7394,7395,7477],{},[76,7396,7397,7411,7423,7429,7431,7433,7435,7437,7449,7451,7453,7455,7457,7459,7461,7463,7465,7467,7473,7475],{},[6523,7398,7399,7401,7409],{},[137,7400,6527],{},[76,7402,7403,7405,7407],{},[79,7404,3659],{},[137,7406,791],{},[473,7408,471],{},[79,7410,2119],{},[76,7412,7413,7415,7421],{},[137,7414,368],{"fence":90},[1174,7416,7417,7419],{"linethickness":6561},[79,7418,2119],{},[79,7420,3659],{},[137,7422,373],{"fence":90},[132,7424,7425,7427],{},[79,7426,3432],{},[79,7428,3659],{},[137,7430,368],{"stretchy":367},[473,7432,637],{},[137,7434,6309],{},[79,7436,3432],{},[132,7438,7439,7441],{},[137,7440,373],{"stretchy":367},[76,7442,7443,7445,7447],{},[79,7444,2119],{},[137,7446,6309],{},[79,7448,3659],{},[137,7450,791],{},[137,7452,368],{"stretchy":367},[79,7454,3432],{},[137,7456,2000],{},[137,7458,368],{"stretchy":367},[473,7460,637],{},[137,7462,6309],{},[79,7464,3432],{},[137,7466,373],{"stretchy":367},[132,7468,7469,7471],{},[137,7470,373],{"stretchy":367},[79,7472,2119],{},[137,7474,791],{},[473,7476,637],{},[83,7478,7479],{"encoding":85},"\\sum_{k=0}^{n}\n\\binom{n}{k}\nx^k(1-x)^{n-k}\n=(x+(1-x))^n\n=1",[59,7481,7483,7663,7719,7740,7761,7812],{"className":7482,"ariaHidden":90},[89],[59,7484,7486,7489,7556,7559,7619,7648,7651,7654,7657,7660],{"className":7485},[94],[59,7487],{"className":7488,"style":6684},[98],[59,7490,7492],{"className":7491},[6688,6689],[59,7493,7495,7548],{"className":7494},[166,527],[59,7496,7498,7545],{"className":7497},[170],[59,7499,7501,7521,7531],{"className":7500,"style":6699},[174],[59,7502,7503,7506],{"style":6702},[59,7504],{"className":7505,"style":6706},[181],[59,7507,7509],{"className":7508},[186,187,188,189],[59,7510,7512,7515,7518],{"className":7511},[103,189],[59,7513,3659],{"className":7514,"style":4157},[103,104,189],[59,7516,791],{"className":7517},[875,189],[59,7519,471],{"className":7520},[103,189],[59,7522,7523,7526],{"style":6724},[59,7524],{"className":7525,"style":6706},[181],[59,7527,7528],{},[59,7529,6527],{"className":7530},[6688,6733,6734],[59,7532,7533,7536],{"style":6737},[59,7534],{"className":7535,"style":6706},[181],[59,7537,7539],{"className":7538},[186,187,188,189],[59,7540,7542],{"className":7541},[103,189],[59,7543,2119],{"className":7544},[103,104,189],[59,7546,564],{"className":7547},[563],[59,7549,7551],{"className":7550},[170],[59,7552,7554],{"className":7553,"style":6759},[174],[59,7555],{},[59,7557],{"className":7558,"style":983},[870],[59,7560,7562,7568,7613],{"className":7561},[103],[59,7563,7565],{"className":7564,"style":509},[393,508],[59,7566,368],{"className":7567},[513,188],[59,7569,7571],{"className":7570},[1174],[59,7572,7574,7605],{"className":7573},[166,527],[59,7575,7577,7602],{"className":7576},[170],[59,7578,7580,7591],{"className":7579,"style":6877},[174],[59,7581,7582,7585],{"style":4109},[59,7583],{"className":7584,"style":541},[181],[59,7586,7588],{"className":7587},[103],[59,7589,3659],{"className":7590,"style":4157},[103,104],[59,7592,7593,7596],{"style":4203},[59,7594],{"className":7595,"style":541},[181],[59,7597,7599],{"className":7598},[103],[59,7600,2119],{"className":7601},[103,104],[59,7603,564],{"className":7604},[563],[59,7606,7608],{"className":7607},[170],[59,7609,7611],{"className":7610,"style":4298},[174],[59,7612],{},[59,7614,7616],{"className":7615,"style":509},[400,508],[59,7617,373],{"className":7618},[513,188],[59,7620,7622,7625],{"className":7621},[103],[59,7623,3432],{"className":7624},[103,104],[59,7626,7628],{"className":7627},[162],[59,7629,7631],{"className":7630},[166],[59,7632,7634],{"className":7633},[170],[59,7635,7637],{"className":7636,"style":6935},[174],[59,7638,7639,7642],{"style":1711},[59,7640],{"className":7641,"style":182},[181],[59,7643,7645],{"className":7644},[186,187,188,189],[59,7646,3659],{"className":7647,"style":4157},[103,104,189],[59,7649,368],{"className":7650},[393],[59,7652,637],{"className":7653},[103],[59,7655],{"className":7656,"style":105},[870],[59,7658,6309],{"className":7659},[193],[59,7661],{"className":7662,"style":105},[870],[59,7664,7666,7669,7672,7710,7713,7716],{"className":7665},[94],[59,7667],{"className":7668,"style":6968},[98],[59,7670,3432],{"className":7671},[103,104],[59,7673,7675,7678],{"className":7674},[400],[59,7676,373],{"className":7677},[400],[59,7679,7681],{"className":7680},[162],[59,7682,7684],{"className":7683},[166],[59,7685,7687],{"className":7686},[170],[59,7688,7690],{"className":7689,"style":6935},[174],[59,7691,7692,7695],{"style":1711},[59,7693],{"className":7694,"style":182},[181],[59,7696,7698],{"className":7697},[186,187,188,189],[59,7699,7701,7704,7707],{"className":7700},[103,189],[59,7702,2119],{"className":7703},[103,104,189],[59,7705,6309],{"class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1。",[14,8453,8454],{},"但我们还可以从概率的角度解读。",[14,8456,8457],{},"设随机变量",[59,8459,8461],{"className":8460,"translate":63},[767],[59,8462,8464,8519],{"className":8463,"translate":63},[62],[59,8465,8467],{"className":8466},[67],[69,8468,8469],{"xmlns":71,"display":776},[73,8470,8471,8516],{},[76,8472,8473,8480,8483,8504,8506,8508,8510,8512,8514],{},[2924,8474,8475,8478],{},[79,8476,8477],{},"X",[79,8479,2119],{},[137,8481,8482],{},"∼",[76,8484,8485,8487,8489,8491,8494,8497,8499,8502],{},[79,8486,6505],{"mathvariant":1091},[79,8488,2933],{"mathvariant":1091},[79,8490,2119],{"mathvariant":1091},[79,8492,8493],{"mathvariant":1091},"o",[79,8495,8496],{"mathvariant":1091},"m",[79,8498,2933],{"mathvariant":1091},[79,8500,8501],{"mathvariant":1091},"a",[79,8503,3714],{"mathvariant":1091},[137,8505,368],{"stretchy":367},[79,8507,2119],{},[137,8509,1345],{"separator":90},[79,8511,3432],{},[137,8513,373],{"stretchy":367},[79,8515,6349],{"mathvariant":1091},[83,8517,8518],{"encoding":85},"X_n\\sim 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",[59,10117,10119,10132],{"className":10118,"translate":63},[62],[59,10120,10122],{"className":10121},[67],[69,10123,10124],{"xmlns":71},[73,10125,10126,10130],{},[76,10127,10128],{},[79,10129,3432],{},[83,10131,3432],{"encoding":85},[59,10133,10135],{"className":10134,"ariaHidden":90},[89],[59,10136,10138,10141],{"className":10137},[94],[59,10139],{"className":10140,"style":310},[98],[59,10142,3432],{"className":10143},[103,104]," 附近。",[14,10146,10147,10148,10201],{},"当 ",[59,10149,10151,10171],{"className":10150,"translate":63},[62],[59,10152,10154],{"className":10153},[67],[69,10155,10156],{"xmlns":71},[73,10157,10158,10168],{},[76,10159,10160,10162,10165],{},[79,10161,2119],{},[137,10163,10164],{},"→",[79,10166,10167],{"mathvariant":1091},"∞",[83,10169,10170],{"encoding":85},"n\\to\\infty",[59,10172,10174,10192],{"className":10173,"ariaHidden":90},[89],[59,10175,10177,10180,10183,10186,10189],{"className":10176},[94],[59,10178],{"className":10179,"style":310},[98],[59,10181,2119],{"className":10182},[103,104],[59,10184],{"className":10185,"style":871},[870],[59,10187,10164],{"className":10188},[875],[59,10190],{"className":10191,"style":871},[870],[59,10193,10195,10198],{"className":10194},[94],[59,10196],{"className":10197,"style":310},[98],[59,10199,10167],{"className":10200},[103]," 时，",[59,10203,10205],{"className":10204,"translate":63},[767],[59,10206,10208,10236],{"className":10207,"translate":63},[62],[59,10209,10211],{"className":10210},[67],[69,10212,10213],{"xmlns":71,"display":776},[73,10214,10215,10233],{},[76,10216,10217,10227,10229,10231],{},[1174,10218,10219,10225],{},[2924,10220,10221,10223],{},[79,10222,8477],{},[79,10224,2119],{},[79,10226,2119],{},[137,10228,10164],{},[79,10230,3432],{},[79,10232,6349],{"mathvariant":1091},[83,10234,10235],{"encoding":85},"\\frac{X_n}{n}\n\\to x.",[59,10237,10239,10353],{"className":10238,"ariaHidden":90},[89],[59,10240,10242,10245,10344,10347,10350],{"className":10241},[94],[59,10243],{"className":10244,"style":10013},[98],[59,10246,10248,10251,10341],{"className":10247},[103],[59,10249],{"className":10250},[393,1222],[59,10252,10254],{"className":10253},[1174],[59,10255,10257,10333],{"className":10256},[166,527],[59,10258,10260,10330],{"className":10259},[170],[59,10261,10263,10274,10282],{"className":10262,"style":9150},[174],[59,10264,10265,10268],{"style":4109},[59,10266],{"className":10267,"style":541},[181],[59,10269,10271],{"className":10270},[103],[59,10272,2119],{"className":10273},[103,104],[59,10275,10276,10279],{"style":1253},[59,10277],{"className":10278,"style":541},[181],[59,10280],{"className":10281,"style":1261},[1260],[59,10283,10284,10287],{"style":4203},[59,10285],{"className":10286,"style":541},[181],[59,10288,10290],{"className":10289},[103],[59,10291,10293,10296],{"className":10292},[103],[59,10294,8477],{"className":10295,"style":8535},[103,104],[59,10297,10299],{"className":10298},[162],[59,10300,10302,10322],{"className":10301},[166,527],[59,10303,10305,10319],{"className":10304},[170],[59,10306,10308],{"className":10307,"style":6627},[174],[59,10309,10310,10313],{"style":8550},[59,10311],{"className":10312,"style":182},[181],[59,10314,10316],{"className":10315},[186,187,188,189],[59,10317,2119],{"className":10318},[103,104,189],[59,10320,564],{"className":10321},[563],[59,10323,10325],{"className":10324},[170],[59,10326,10328],{"className":10327,"style":3092},[174],[59,10329],{},[59,10331,564],{"className":10332},[563],[59,10334,10336],{"className":10335},[170],[59,10337,10339],{"className":10338,"style":4298},[174],[59,10340],{},[59,10342],{"className":10343},[400,1222],[59,10345],{"className":10346,"style":871},[870],[59,10348,10164],{"className":10349},[875],[59,10351],{"className":10352,"style":871},[870],[59,10354,10356,10359,10362],{"className":10355},[94],[59,10357],{"className":10358,"style":310},[98],[59,10360,3432],{"className":10361},[103,104],[59,10363,6349],{"className":10364},[103],[14,10366,10367],{},"于是",[59,10369,10371],{"className":10370,"translate":63},[767],[59,10372,10374,10419],{"className":10373,"translate":63},[62],[59,10375,10377],{"className":10376},[67],[69,10378,10379],{"xmlns":71,"display":776},[73,10380,10381,10416],{},[76,10382,10383,10385,10387,10403,10406,10408,10410,10412,10414],{},[79,10384,6010],{},[1991,10386,6542],{},[76,10388,10389,10391,10401],{},[137,10390,368],{"fence":90},[1174,10392,10393,10399],{},[2924,10394,10395,10397],{},[79,10396,8477],{},[79,10398,2119],{},[79,10400,2119],{},[137,10402,373],{"fence":90},[137,10404,10405],{},"≈",[79,10407,6010],{},[137,10409,368],{"stretchy":367},[79,10411,3432],{},[137,10413,373],{"stretchy":367},[79,10415,6349],{"mathvariant":1091},[83,10417,10418],{"encoding":85},"f\\!\\left(\\frac{X_n}{n}\\right)\n\\approx\nf(x).",[59,10420,10422,10560],{"className":10421,"ariaHidden":90},[89],[59,10423,10425,10428,10431,10434,10437,10551,10554,10557],{"className":10424},[94],[59,10426],{"className":10427,"style":500},[98],[59,10429,6010],{"className":10430,"style":6047},[103,104],[59,10432],{"className":10433,"style":6771},[870],[59,10435],{"className":10436,"style":983},[870],[59,10438,10440,10446,10545],{"className":10439},[504],[59,10441,10443],{"className":10442,"style":509},[393,508],[59,10444,368],{"className":10445},[513,188],[59,10447,10449,10452,10542],{"className":10448},[103],[59,10450],{"className":10451},[393,1222],[59,10453,10455],{"className":10454},[1174],[59,10456,10458,10534],{"className":10457},[166,527],[59,10459,10461,10531],{"className":10460},[170],[59,10462,10464,10475,10483],{"className":10463,"style":9150},[174],[59,10465,10466,10469],{"style":4109},[59,10467],{"className":10468,"style":541},[181],[59,10470,10472],{"className":10471},[103],[59,10473,2119],{"className":10474},[103,104],[59,10476,10477,10480],{"style":1253},[59,10478],{"className":10479,"style":541},[181],[59,10481],{"className":10482,"style":1261},[1260],[59,10484,10485,10488],{"style":4203},[59,10486],{"className":10487,"style":541},[181],[59,10489,10491],{"className":10490},[103],[59,10492,10494,10497],{"className":10493},[103],[59,10495,8477],{"className":10496,"style":8535},[103,104],[59,10498,10500],{"className":10499},[162],[59,10501,10503,10523],{"className":10502},[166,527],[59,10504,10506,10520],{"className":10505},[170],[59,10507,10509],{"className":10508,"style":6627},[174],[59,10510,10511,10514],{"style":8550},[59,10512],{"className":10513,"style":182},[181],[59,10515,10517],{"className":10516},[186,187,188,189],[59,10518,2119],{"className":10519},[103,104,189],[59,10521,564],{"className":10522},[563],[59,10524,10526],{"className":10525},[170],[59,10527,10529],{"className":10528,"style":3092},[174],[59,10530],{},[59,10532,564],{"className":10533},[563],[59,10535,10537],{"className":10536},[170],[59,10538,10540],{"className":10539,"style":4298},[174],[59,10541],{},[59,10543],{"className":10544},[400,1222],[59,10546,10548],{"className":10547,"style":509},[400,508],[59,10549,373],{"className":10550},[513,188],[59,10552],{"className":10553,"style":871},[870],[59,10555,10405],{"className":10556},[875],[59,10558],{"className":10559,"style":871},[870],[59,10561,10563,10566,10569,10572,10575,10578],{"className":10562},[94],[59,10564],{"className":10565,"style":386},[98],[59,10567,6010],{"className":10568,"style":6047},[103,104],[59,10570,368],{"className":10571},[393],[59,10573,3432],{"className":10574},[103,104],[59,10576,373],{"className":10577},[400],[59,10579,6349],{"className":10580},[103],[14,10582,7269],{},[59,10584,10586],{"className":10585,"translate":63},[767],[59,10587,10589,10663],{"className":10588,"translate":63},[62],[59,10590,10592],{"className":10591},[67],[69,10593,10594],{"xmlns":71,"display":776},[73,10595,10596,10660],{},[76,10597,10598,10604,10606,10608,10610,10612,10614,10616,10618,10620,10622,10648,10650,10652,10654,10656,10658],{},[2924,10599,10600,10602],{},[79,10601,6505],{},[79,10603,2119],{},[137,10605,368],{"stretchy":367},[79,10607,6010],{},[137,10609,373],{"stretchy":367},[137,10611,368],{"stretchy":367},[79,10613,3432],{},[137,10615,373],{"stretchy":367},[137,10617,791],{},[79,10619,8979],{},[1991,10621,6542],{},[76,10623,10624,10626,10628,10630,10646],{},[137,10625,455],{"fence":90},[79,10627,6010],{},[1991,10629,6542],{},[76,10631,10632,10634,10644],{},[137,10633,368],{"fence":90},[1174,10635,10636,10642],{},[2924,10637,10638,10640],{},[79,10639,8477],{},[79,10641,2119],{},[79,10643,2119],{},[137,10645,373],{"fence":90},[137,10647,487],{"fence":90},[137,10649,10405],{},[79,10651,6010],{},[137,10653,368],{"stretchy":367},[79,10655,3432],{},[137,10657,373],{"stretchy":367},[79,10659,6349],{"mathvariant":1091},[83,10661,10662],{"encoding":85},"B_n(f)(x)\n=\nE\\!\\left[\nf\\!\\left(\\frac{X_n}{n}\\right)\n\\right]\n\\approx\nf(x).",[59,10664,10666,10739,10901],{"className":10665,"ariaHidden":90},[89],[59,10667,10669,10672,10712,10715,10718,10721,10724,10727,10730,10733,10736],{"className":10668},[94],[59,10670],{"className":10671,"style":386},[98],[59,10673,10675,10678],{"className":10674},[103],[59,10676,6505],{"className":10677,"style":6614},[103,104],[59,10679,10681],{"className":10680},[162],[59,10682,10684,10704],{"className":10683},[166,527],[59,10685,10687,10701],{"className":10686},[170],[59,10688,10690],{"className":10689,"style":6627},[174],[59,10691,10692,10695],{"style":6630},[59,10693],{"className":10694,"style":182},[181],[59,10696,10698],{"className":10697},[186,187,188,189],[59,10699,2119],{"className":10700},[103,104,189],[59,10702,564],{"className":10703},[563],[59,10705,10707],{"className":10706},[170],[59,10708,10710],{"className":10709,"style":3092},[174],[59,10711],{},[59,10713,368],{"className":10714},[393],[59,10716,6010],{"className":10717,"style":6047},[103,104],[59,10719,373],{"className":10720},[400],[59,10722,368],{"className":10723},[393],[59,10725,3432],{"className":10726},[103,104],[59,10728,373],{"className":10729},[400],[59,10731],{"className":10732,"style":871},[870],[59,10734,791],{"className":10735},[875],[59,10737],{"className":10738,"style":871},[870],[59,10740,10742,10745,10748,10751,10754,10892,10895,10898],{"className":10741},[94],[59,10743],{"className":10744,"style":500},[98],[59,10746,8979],{"className":10747,"style":9098},[103,104],[59,10749],{"className":10750,"style":6771},[870],[59,10752],{"className":10753,"style":983},[870],[59,10755,10757,10763,10766,10769,10772,10886],{"className":10756},[504],[59,10758,10760],{"className":10759,"style":509},[393,508],[59,10761,455],{"className":10762},[513,188],[59,10764,6010],{"className":10765,"style":6047},[103,104],[59,10767],{"className":10768,"style":6771},[870],[59,10770],{"className":10771,"style":983},[870],[59,10773,10775,10781,10880],{"className":10774},[504],[59,10776,10778],{"className":10777,"style":509},[393,508],[59,10779,368],{"className":10780},[513,188],[59,10782,10784,10787,10877],{"className":10783},[103],[59,10785],{"className":10786},[393,1222],[59,10788,10790],{"className":10789},[1174],[59,10791,10793,10869],{"className":10792},[166,527],[59,10794,10796,10866],{"className":10795},[170],[59,10797,10799,10810,10818],{"className":10798,"style":9150},[174],[59,10800,10801,10804],{"style":4109},[59,10802],{"className":10803,"style":541},[181],[59,10805,10807],{"className":10806},[103],[59,10808,2119],{"className":10809},[103,104],[59,10811,10812,10815],{"style":1253},[59,10813],{"className":10814,"style":541},[181],[59,10816],{"className":10817,"style":1261},[1260],[59,10819,10820,10823],{"style":4203},[59,10821],{"className":10822,"style":541},[181],[59,10824,10826],{"className":10825},[103],[59,10827,10829,10832],{"className":10828},[103],[59,10830,8477],{"className":10831,"style":8535},[103,104],[59,10833,10835],{"className":10834},[162],[59,10836,10838,10858],{"className":10837},[166,527],[59,10839,10841,10855],{"className":10840},[170],[59,10842,10844],{"className":10843,"style":6627},[174],[59,10845,10846,10849],{"style":8550},[59,10847],{"className":10848,"style":182},[181],[59,10850,10852],{"className":10851},[186,187,188,189],[59,10853,2119],{"className":10854},[103,104,189],[59,10856,564],{"className":10857},[563],[59,10859,10861],{"className":10860},[170],[59,10862,10864],{"className":10863,"style":3092},[174],[59,10865],{},[59,10867,564],{"className":10868},[563],[59,10870,10872],{"className":10871},[170],[59,10873,10875],{"className":10874,"style":4298},[174],[59,10876],{},[59,10878],{"className":10879},[400,1222],[59,10881,10883],{"className":10882,"style":509},[400,508],[59,10884,373],{"className":10885},[513,188],[59,10887,10889],{"className":10888,"style":509},[400,508],[59,10890,487],{"className":10891},[513,188],[59,10893],{"className":10894,"style":871},[870],[59,10896,10405],{"className":10897},[875],[59,10899],{"className":10900,"style":871},[870],[59,10902,10904,10907,10910,10913,10916,10919],{"className":10903},[94],[59,10905],{"className":10906,"style":386},[98],[59,10908,6010],{"className":10909,"style":6047},[103,104],[59,10911,368],{"className":10912},[393],[59,10914,3432],{"className":10915},[103,104],[59,10917,373],{"className":10918},[400],[59,10920,6349],{"className":10921},[103],[14,10923,10924],{},"直觉上，这成立。",[5987,10926,10927],{"id":10927},"严格证明",[14,10929,10930,10931,10959,10960,10988,10989,11039,11040,11094],{},"由于 ",[59,10932,10934,10947],{"className":10933,"translate":63},[62],[59,10935,10937],{"className":10936},[67],[69,10938,10939],{"xmlns":71},[73,10940,10941,10945],{},[76,10942,10943],{},[79,10944,6010],{},[83,10946,6010],{"encoding":85},[59,10948,10950],{"className":10949,"ariaHidden":90},[89],[59,10951,10953,10956],{"className":10952},[94],[59,10954],{"className":10955,"style":6043},[98],[59,10957,6010],{"className":10958,"style":6047},[103,104]," 在闭区间上连续，由闭区间上连续函数的性质：",[59,10961,10963,10976],{"className":10962,"translate":63},[62],[59,10964,10966],{"className":10965},[67],[69,10967,10968],{"xmlns":71},[73,10969,10970,10974],{},[76,10971,10972],{},[79,10973,6010],{},[83,10975,6010],{"encoding":85},[59,10977,10979],{"className":10978,"ariaHidden":90},[89],[59,10980,10982,10985],{"className":10981},[94],[59,10983],{"className":10984,"style":6043},[98],[59,10986,6010],{"className":10987,"style":6047},[103,104]," 必一致连续。因此，对任意 ",[59,10990,10992,11009],{"className":10991,"translate":63},[62],[59,10993,10995],{"className":10994},[67],[69,10996,10997],{"xmlns":71},[73,10998,10999,11007],{},[76,11000,11001,11003,11005],{},[79,11002,6192],{},[137,11004,6195],{},[473,11006,471],{},[83,11008,6200],{"encoding":85},[59,11010,11012,11030],{"className":11011,"ariaHidden":90},[89],[59,11013,11015,11018,11021,11024,11027],{"className":11014},[94],[59,11016],{"className":11017,"style":6210},[98],[59,11019,6192],{"className":11020},[103,104],[59,11022],{"className":11023,"style":871},[870],[59,11025,6195],{"className":11026},[875],[59,11028],{"className":11029,"style":871},[870],[59,11031,11033,11036],{"className":11032},[94],[59,11034],{"className":11035,"style":1065},[98],[59,11037,471],{"className":11038},[103]," 存在 ",[59,11041,11043,11062],{"className":11042,"translate":63},[62],[59,11044,11046],{"className":11045},[67],[69,11047,11048],{"xmlns":71},[73,11049,11050,11059],{},[76,11051,11052,11055,11057],{},[79,11053,11054],{},"δ",[137,11056,6195],{},[473,11058,471],{},[83,11060,11061],{"encoding":85},"\\delta>0",[59,11063,11065,11085],{"className":11064,"ariaHidden":90},[89],[59,11066,11068,11072,11076,11079,11082],{"className":11067},[94],[59,11069],{"className":11070,"style":11071},[98],"height:0.7335em;vertical-align:-0.0391em;",[59,11073,11054],{"className":11074,"style":11075},[103,104],"margin-right:0.0379em;",[59,11077],{"className":11078,"style":871},[870],[59,11080,6195],{"className":11081},[875],[59,11083],{"className":11084,"style":871},[870],[59,11086,11088,11091],{"className":11087},[94],[59,11089],{"className":11090,"style":1065},[98],[59,11092,471],{"className":11093},[103],"，使得只要",[59,11096,11098],{"className":11097,"translate":63},[767],[59,11099,11101,11128],{"className":11100,"translate":63},[62],[59,11102,11104],{"className":11103},[67],[69,11105,11106],{"xmlns":71,"display":776},[73,11107,11108,11125],{},[76,11109,11110,11112,11115,11117,11119,11121,11123],{},[79,11111,6298],{"mathvariant":1091},[79,11113,11114],{},"u",[137,11116,6309],{},[79,11118,331],{},[79,11120,6298],{"mathvariant":1091},[137,11122,6322],{},[79,11124,11054],{},[83,11126,11127],{"encoding":85},"|u-v|\u003C\\delta",[59,11129,11131,11152,11173],{"className":11130,"ariaHidden":90},[89],[59,11132,11134,11137,11140,11143,11146,11149],{"className":11133},[94],[59,11135],{"className":11136,"style":386},[98],[59,11138,6298],{"className":11139},[103],[59,11141,11114],{"className":11142},[103,104],[59,11144],{"className":11145,"style":105},[870],[59,11147,6309],{"className":11148},[193],[59,11150],{"className":11151,"style":105},[870],[59,11153,11155,11158,11161,11164,11167,11170],{"className":11154},[94],[59,11156],{"className":11157,"style":386},[98],[59,11159,331],{"className":11160,"style":314},[103,104],[59,11162,6298],{"className":11163},[103],[59,11165],{"className":11166,"style":871},[870],[59,11168,6322],{"className":11169},[875],[59,11171],{"className":11172,"style":871},[870],[59,11174,11176,11180],{"className":11175},[94],[59,11177],{"className":11178,"style":11179},[98],"height:0.6944em;",[59,11181,11054],{"className":11182,"style":11075},[103,104],[14,11184,11185],{},"就有",[59,11187,11189],{"className":11188,"translate":63},[767],[59,11190,11192,11236],{"className":11191,"translate":63},[62],[59,11193,11195],{"className":11194},[67],[69,11196,11197],{"xmlns":71,"display":776},[73,11198,11199,11233],{},[76,11200,11201,11203,11205,11207,11209,11211,11213,11215,11217,11219,11221,11223,11225,11231],{},[79,11202,6298],{"mathvariant":1091},[79,11204,6010],{},[137,11206,368],{"stretchy":367},[79,11208,11114],{},[137,11210,373],{"stretchy":367},[137,11212,6309],{},[79,11214,6010],{},[137,11216,368],{"stretchy":367},[79,11218,331],{},[137,11220,373],{"stretchy":367},[79,11222,6298],{"mathvariant":1091},[137,11224,6322],{},[1174,11226,11227,11229],{},[79,11228,6192],{},[473,11230,475],{},[79,11232,6349],{"mathvariant":1091},[83,11234,11235],{"encoding":85},"|f(u)-f(v)|\u003C\\frac{\\varepsilon}{2}.",[59,11237,11239,11269,11299],{"className":11238,"ariaHidden":90},[89],[59,11240,11242,11245,11248,11251,11254,11257,11260,11263,11266],{"className":11241},[94],[59,11243],{"className":11244,"style":386},[98],[59,11246,6298],{"className":11247},[103],[59,11249,6010],{"className":11250,"style":6047},[103,104],[59,11252,368],{"className":11253},[393],[59,11255,11114],{"className":11256},[103,104],[59,11258,373],{"className":11259},[400],[59,11261],{"className":11262,"style":105},[870],[59,11264,6309],{"className":11265},[193],[59,11267],{"className":11268,"style":105},[870],[59,11270,11272,11275,11278,11281,11284,11287,11290,11293,11296],{"className":11271},[94],[59,11273],{"className":11274,"style":386},[98],[59,11276,6010],{"className":11277,"style":6047},[103,104],[59,11279,368],{"className":11280},[393],[59,11282,331],{"className":11283,"style":314},[103,104],[59,11285,373],{"className":11286},[400],[59,11288,6298],{"className":11289},[103],[59,11291],{"className":11292,"style":871},[870],[59,11294,6322],{"className":11295},[875],[59,11297],{"className":11298,"style":871},[870],[59,11300,11302,11306,11368],{"className":11301},[94],[59,11303],{"className":11304,"style":11305},[98],"height:1.7936em;vertical-align:-0.686em;",[59,11307,11309,11312,11365],{"className":11308},[103],[59,11310],{"className":11311},[393,1222],[59,11313,11315],{"className":11314},[1174],[59,11316,11318,11357],{"className":11317},[166,527],[59,11319,11321,11354],{"className":11320},[170],[59,11322,11324,11335,11343],{"className":11323,"style":6877},[174],[59,11325,11326,11329],{"style":4109},[59,11327],{"className":11328,"style":541},[181],[59,11330,11332],{"className":11331},[103],[59,11333,475],{"className":11334},[103],[59,11336,11337,11340],{"style":1253},[59,11338],{"className":11339,"style":541},[181],[59,11341],{"className":11342,"style":1261},[1260],[59,11344,11345,11348],{"style":4203},[59,11346],{"className":11347,"style":541},[181],[59,11349,11351],{"className":11350},[103],[59,11352,6192],{"className":11353},[103,104],[59,11355,564],{"className":11356},[563],[59,11358,11360],{"className":11359},[170],[59,11361,11363],{"className":11362,"style":4298},[174],[59,11364],{},[59,11366],{"className":11367},[400,1222],[59,11369,6349],{"className":11370},[103],[14,11372,11373],{},"分解误差：",[59,11375,11377],{"className":11376,"translate":63},[767],[59,11378,11380,11467],{"className":11379,"translate":63},[62],[59,11381,11383],{"className":11382},[67],[69,11384,11385],{"xmlns":71,"display":776},[73,11386,11387,11464],{},[76,11388,11389,11391,11397,11399,11401,11403,11405,11407,11409,11411,11413,11415,11417,11419,11421,11423,11462],{},[79,11390,6298],{"mathvariant":1091},[2924,11392,11393,11395],{},[79,11394,6505],{},[79,11396,2119],{},[137,11398,368],{"stretchy":367},[79,11400,6010],{},[137,11402,373],{"stretchy":367},[137,11404,368],{"stretchy":367},[79,11406,3432],{},[137,11408,373],{"stretchy":367},[137,11410,6309],{},[79,11412,6010],{},[137,11414,368],{"stretchy":367},[79,11416,3432],{},[137,11418,373],{"stretchy":367},[79,11420,6298],{"mathvariant":1091},[137,11422,791],{},[76,11424,11425,11427,11429,11432,11434,11436,11442,11444,11446,11448,11450,11452,11454,11456,11458,11460],{},[137,11426,6298],{"fence":90},[79,11428,8979],{},[137,11430,368],{"fence":367,"stretchy":90,"minsize":11431,"maxsize":11431},"1.8em",[79,11433,6010],{},[137,11435,368],{"stretchy":367},[2924,11437,11438,11440],{},[79,11439,8477],{},[79,11441,2119],{},[79,11443,1092],{"mathvariant":1091},[79,11445,2119],{},[137,11447,373],{"stretchy":367},[137,11449,6309],{},[79,11451,6010],{},[137,11453,368],{"stretchy":367},[79,11455,3432],{},[137,11457,373],{"stretchy":367},[137,11459,373],{"fence":367,"stretchy":90,"minsize":11431,"maxsize":11431},[137,11461,6298],{"fence":90},[79,11463,6349],{"mathvariant":1091},[83,11465,11466],{"encoding":85},"|B_n(f)(x)-f(x)|\n=\n\n\\left|\nE\\Big(\nf(X_n\u002Fn)-f(x)\n\\Big)\n\\right|.",[59,11468,11470,11546,11576],{"className":11469,"ariaHidden":90},[89],[59,11471,11473,11476,11479,11519,11522,11525,11528,11531,11534,11537,11540,11543],{"className":11472},[94],[59,11474],{"className":11475,"style":386},[98],[59,11477,6298],{"className":11478},[103],[59,11480,11482,11485],{"className":11481},[103],[59,11483,6505],{"className":11484,"style":6614},[103,104],[59,11486,11488],{"className":11487},[162],[59,11489,11491,11511],{"className":11490},[166,527],[59,11492,11494,11508],{"className":11493},[170],[59,11495,11497],{"className":11496,"style":6627},[174],[59,11498,11499,11502],{"style":6630},[59,11500],{"className":11501,"style":182},[181],[59,11503,11505],{"className":11504},[186,187,188,189],[59,11506,2119],{"className":11507},[103,104,189],[59,11509,564],{"className":11510},[563],[59,11512,11514],{"className":11513},[170],[59,11515,11517],{"className":11516,"style":3092},[174],[59,11518],{},[59,11520,368],{"className":11521},[393],[59,11523,6010],{"className":11524,"style":6047},[103,104],[59,11526,373],{"className":11527},[400],[59,11529,368],{"className":11530},[393],[59,11532,3432],{"className":11533},[103,104],[59,11535,373],{"className":11536},[400],[59,11538],{"className":11539,"style":105},[870],[59,11541,6309],{"className":11542},[193],[59,11544],{"className":11545,"style":105},[870],[59,11547,11549,11552,11555,11558,11561,11564,11567,11570,11573],{"className":11548},[94],[59,11550],{"className":11551,"style":386},[98],[59,11553,6010],{"className":11554,"style":6047},[103,104],[59,11556,368],{"className":11557},[393],[59,11559,3432],{"className":11560},[103,104],[59,11562,373],{"className":11563},[400],[59,11565,6298],{"className":11566},[103],[59,11568],{"className":11569,"style":871},[870],[59,11571,791],{"className":11572},[875],[59,11574],{"className":11575,"style":871},[870],[59,11577,11579,11583,11761,11764],{"className":11578},[94],[59,11580],{"className":11581,"style":11582},[98],"height:1.8em;vertical-align:-0.65em;",[59,11584,11586,11632,11635,11642,11645,11648,11688,11691,11694,11697,11700,11703,11706,11709,11712,11715,11718,11724],{"className":11585},[504],[59,11587,11589],{"className":11588},[393],[59,11590,11593],{"className":11591},[513,11592],"mult",[59,11594,11596,11623],{"className":11595},[166,527],[59,11597,11599,11620],{"className":11598},[170],[59,11600,11603],{"className":11601,"style":11602},[174],"height:1.15em;",[59,11604,11606,11610],{"style":11605},"top:-3.15em;",[59,11607],{"className":11608,"style":11609},[181],"height:3.8em;",[59,11611,11613],{"style":11612},"width:0.333em;height:1.8em;",[1652,11614,11617],{"xmlns":1654,"width":11615,"height":11431,"viewBox":11616},"0.333em","0 0 333 1800",[1660,11618],{"d":11619},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[59,11621,564],{"className":11622},[563],[59,11624,11626],{"className":11625},[170],[59,11627,11630],{"className":11628,"style":11629},[174],"height:0.65em;",[59,11631],{},[59,11633,8979],{"className":11634,"style":9098},[103,104],[59,11636,11638],{"className":11637},[103],[59,11639,368],{"className":11640},[513,11641],"size2",[59,11643,6010],{"className":11644,"style":6047},[103,104],[59,11646,368],{"className":11647},[393],[59,11649,11651,11654],{"className":11650},[103],[59,11652,8477],{"className":11653,"style":8535},[103,104],[59,11655,11657],{"className":11656},[162],[59,11658,11660,11680],{"className":11659},[166,527],[59,11661,11663,11677],{"className":11662},[170],[59,11664,11666],{"className":11665,"style":6627},[174],[59,11667,11668,11671],{"style":8550},[59,11669],{"className":11670,"style":182},[181],[59,11672,11674],{"className":11673},[186,187,188,189],[59,11675,2119],{"className":11676},[103,104,189],[59,11678,564],{"className":11679},[563],[59,11681,11683],{"className":11682},[170],[59,11684,11686],{"className":11685,"style":3092},[174],[59,11687],{},[59,11689,1092],{"className":11690},[103],[59,11692,2119],{"className":11693},[103,104],[59,11695,373],{"className":11696},[400],[59,11698],{"className":11699,"style":105},[870],[59,11701,6309],{"className":11702},[193],[59,11704],{"className":11705,"style":105},[870],[59,11707,6010],{"className":11708,"style":6047},[103,104],[59,11710,368],{"className":11711},[393],[59,11713,3432],{"className":11714},[103,104],[59,11716,373],{"className":11717},[400],[59,11719,11721],{"className":11720},[103],[59,11722,373],{"className":11723},[513,11641],[59,11725,11727],{"className":11726},[400],[59,11728,11730],{"className":11729},[513,11592],[59,11731,11733,11753],{"className":11732},[166,527],[59,11734,11736,11750],{"className":11735},[170],[59,11737,11739],{"className":11738,"style":11602},[174],[59,11740,11741,11744],{"style":11605},[59,11742],{"className":11743,"style":11609},[181],[59,11745,11746],{"style":11612},[1652,11747,11748],{"xmlns":1654,"width":11615,"height":11431,"viewBox":11616},[1660,11749],{"d":11619},[59,11751,564],{"className":11752},[563],[59,11754,11756],{"className":11755},[170],[59,11757,11759],{"className":11758,"style":11629},[174],[59,11760],{},[59,11762],{"className":11763,"style":983},[870],[59,11765,6349],{"className":11766},[103],[14,11768,11769],{},"利用绝对值期望不等式：",[59,11771,11773],{"className":11772,"translate":63},[767],[59,11774,11776,11829],{"className":11775,"translate":63},[62],[59,11777,11779],{"className":11778},[67],[69,11780,11781],{"xmlns":71,"display":776},[73,11782,11783,11826],{},[76,11784,11785,11788,11790,11792,11794,11796,11798,11804,11806,11808,11810,11812,11814,11816,11818,11820,11822,11824],{},[137,11786,11787],{},"≤",[79,11789,8979],{},[137,11791,455],{"fence":367,"stretchy":90,"minsize":11431,"maxsize":11431},[79,11793,6298],{"mathvariant":1091},[79,11795,6010],{},[137,11797,368],{"stretchy":367},[2924,11799,11800,11802],{},[79,11801,8477],{},[79,11803,2119],{},[79,11805,1092],{"mathvariant":1091},[79,11807,2119],{},[137,11809,373],{"stretchy":367},[137,11811,6309],{},[79,11813,6010],{},[137,11815,368],{"stretchy":367},[79,11817,3432],{},[137,11819,373],{"stretchy":367},[79,11821,6298],{"mathvariant":1091},[137,11823,487],{"fence":367,"stretchy":90,"minsize":11431,"maxsize":11431},[79,11825,6349],{"mathvariant":1091},[83,11827,11828],{"encoding":85},"\\le\nE\\Big[\n|f(X_n\u002Fn)-f(x)|\n\\Big].",[59,11830,11832,11845,11927],{"className":11831,"ariaHidden":90},[89],[59,11833,11835,11839,11842],{"className":11834},[94],[59,11836],{"className":11837,"style":11838},[98],"height:0.7719em;vertical-align:-0.136em;",[59,11840,11787],{"className":11841},[875],[59,11843],{"className":11844,"style":871},[870],[59,11846,11848,11851,11854,11860,11863,11866,11869,11909,11912,11915,11918,11921,11924],{"className":11847},[94],[59,11849],{"className":11850,"style":11582},[98],[59,11852,8979],{"className":11853,"style":9098},[103,104],[59,11855,11857],{"className":11856},[103],[59,11858,455],{"className":11859},[513,11641],[59,11861,6298],{"className":11862},[103],[59,11864,6010],{"className":11865,"style":6047},[103,104],[59,11867,368],{"className":11868},[393],[59,11870,11872,11875],{"className":11871},[103],[59,11873,8477],{"className":11874,"style":8535},[103,104],[59,11876,11878],{"className":11877},[162],[59,11879,11881,11901],{"className":11880},[166,527],[59,11882,11884,11898],{"className":11883},[170],[59,11885,11887],{"className":11886,"style":6627},[174],[59,11888,11889,11892],{"style":8550},[59,11890],{"className":11891,"style":182},[181],[59,11893,11895],{"className":11894},[186,187,188,189],[59,11896,2119],{"className":11897},[103,104,189],[59,11899,564],{"className":11900},[563],[59,11902,11904],{"className":11903},[170],[59,11905,11907],{"className":11906,"style":3092},[174],[59,11908],{},[59,11910,1092],{"className":11911},[103],[59,11913,2119],{"className":11914},[103,104],[59,11916,373],{"className":11917},[400],[59,11919],{"className":11920,"style":105},[870],[59,11922,6309],{"className":11923},[193],[59,11925],{"className":11926,"style":105},[870],[59,11928,11930,11933,11936,11939,11942,11945,11948,11954],{"className":11929},[94],[59,11931],{"className":11932,"style":11582},[98],[59,11934,6010],{"className":11935,"style":6047},[103,104],[59,11937,368],{"className":11938},[393],[59,11940,3432],{"className":11941},[103,104],[59,11943,373],{"className":11944},[400],[59,11946,6298],{"className":11947},[103],[59,11949,11951],{"className":11950},[103],[59,11952,487],{"className":11953},[513,11641],[59,11955,6349],{"className":11956},[103],[14,11958,11959],{},"分成两部分：",[59,11961,11963],{"className":11962,"translate":63},[767],[59,11964,11966,12007],{"className":11965,"translate":63},[62],[59,11967,11969],{"className":11968},[67],[69,11970,11971],{"xmlns":71,"display":776},[73,11972,11973,12004],{},[76,11974,11975,11978,11980],{},[79,11976,11977],{},"A",[137,11979,791],{},[76,11981,11982,11984,11990,11992,11994,11996,11998,12000,12002],{},[79,11983,6298],{"mathvariant":1091},[2924,11985,11986,11988],{},[79,11987,8477],{},[79,11989,2119],{},[79,11991,1092],{"mathvariant":1091},[79,11993,2119],{},[137,11995,6309],{},[79,11997,3432],{},[79,11999,6298],{"mathvariant":1091},[137,12001,6322],{},[79,12003,11054],{},[83,12005,12006],{"encoding":85},"A=\n{|X_n\u002Fn-x|\u003C\\delta}",[59,12008,12010,12028],{"className":12009,"ariaHidden":90},[89],[59,12011,12013,12016,12019,12022,12025],{"className":12012},[94],[59,12014],{"className":12015,"style":99},[98],[59,12017,11977],{"className":12018},[103,104],[59,12020],{"className":12021,"style":871},[870],[59,12023,791],{"className":12024},[875],[59,12026],{"className":12027,"style":871},[870],[59,12029,12031,12034],{"className":12030},[94],[59,12032],{"className":12033,"style":386},[98],[59,12035,12037,12040,12080,12083,12086,12089,12092,12095,12098,12101,12104,12107,12110],{"className":12036},[103],[59,12038,6298],{"className":12039},[103],[59,12041,12043,12046],{"className":12042},[103],[59,12044,8477],{"className":12045,"style":8535},[103,104],[59,12047,12049],{"className":12048},[162],[59,12050,12052,12072],{"className":12051},[166,527],[59,12053,12055,12069],{"className":12054},[170],[59,12056,12058],{"className":12057,"style":6627},[174],[59,12059,12060,12063],{"style":8550},[59,12061],{"className":12062,"style":182},[181],[59,12064,12066],{"className":12065},[186,187,188,189],[59,12067,2119],{"className":12068},[103,104,189],[59,12070,564],{"className":12071},[563],[59,12073,12075],{"className":12074},[170],[59,12076,12078],{"className":12077,"style":3092},[174],[59,12079],{},[59,12081,1092],{"className":12082},[103],[59,12084,2119],{"className":12085},[103,104],[59,12087],{"className":12088,"style":105},[870],[59,12090,6309],{"className":12091},[193],[59,12093],{"className":12094,"style":105},[870],[59,12096,3432],{"className":12097},[103,104],[59,12099,6298],{"className":12100},[103],[59,12102],{"className":12103,"style":871},[870],[59,12105,6322],{"className":12106},[875],[59,12108],{"className":12109,"style":871},[870],[59,12111,11054],{"className":12112,"style":11075},[103,104],[14,12114,12115],{},"和",[59,12117,12119],{"className":12118,"translate":63},[767],[59,12120,12122,12170],{"className":12121,"translate":63},[62],[59,12123,12125],{"className":12124},[67],[69,12126,12127],{"xmlns":71,"display":776},[73,12128,12129,12167],{},[76,12130,12131,12138,12140,12165],{},[132,12132,12133,12135],{},[79,12134,11977],{},[79,12136,12137],{},"c",[137,12139,791],{},[76,12141,12142,12144,12150,12152,12154,12156,12158,12160,12163],{},[79,12143,6298],{"mathvariant":1091},[2924,12145,12146,12148],{},[79,12147,8477],{},[79,12149,2119],{},[79,12151,1092],{"mathvariant":1091},[79,12153,2119],{},[137,12155,6309],{},[79,12157,3432],{},[79,12159,6298],{"mathvariant":1091},[137,12161,12162],{},"≥",[79,12164,11054],{},[79,12166,6349],{"mathvariant":1091},[83,12168,12169],{"encoding":85},"A^c=\n{|X_n\u002Fn-x|\\ge \\delta}.",[59,12171,12173,12217],{"className":12172,"ariaHidden":90},[89],[59,12174,12176,12179,12208,12211,12214],{"className":12175},[94],[59,12177],{"className":12178,"style":7791},[98],[59,12180,12182,12185],{"className":12181},[103],[59,12183,11977],{"className":12184},[103,104],[59,12186,12188],{"className":12187},[162],[59,12189,12191],{"className":12190},[166],[59,12192,12194],{"className":12193},[170],[59,12195,12197],{"className":12196,"style":7791},[174],[59,12198,12199,12202],{"style":1711},[59,12200],{"className":12201,"style":182},[181],[59,12203,12205],{"className":12204},[186,187,188,189],[59,12206,12137],{"className":12207},[103,104,189],[59,12209],{"className":12210,"style":871},[870],[59,12212,791],{"className":12213},[875],[59,12215],{"className":12216,"style":871},[870],[59,12218,12220,12223,12302],{"className":12219},[94],[59,12221],{"className":12222,"style":386},[98],[59,12224,12226,12229,12269,12272,12275,12278,12281,12284,12287,12290,12293,12296,12299],{"className":12225},[103],[59,12227,6298],{"className":12228},[103],[59,12230,12232,12235],{"className":12231},[103],[59,12233,8477],{"className":12234,"style":8535},[103,104],[59,12236,12238],{"className":12237},[162],[59,12239,12241,12261],{"className":12240},[166,527],[59,12242,12244,12258],{"className":12243},[170],[59,12245,12247],{"className":12246,"style":6627},[174],[59,12248,12249,12252],{"style":8550},[59,12250],{"className":12251,"style":182},[181],[59,12253,12255],{"className":12254},[186,187,188,189],[59,12256,2119],{"className":12257},[103,104,189],[59,12259,564],{"className":12260},[563],[59,12262,12264],{"className":12263},[170],[59,12265,12267],{"className":12266,"style":3092},[174],[59,12268],{},[59,12270,1092],{"className":12271},[103],[59,12273,2119],{"className":12274},[103,104],[59,12276],{"className":12277,"style":105},[870],[59,12279,6309],{"className":12280},[193],[59,12282],{"className":12283,"style":105},[870],[59,12285,3432],{"className":12286},[103,104],[59,12288,6298],{"className":12289},[103],[59,12291],{"className":12292,"style":871},[870],[59,12294,12162],{"className":12295},[875],[59,12297],{"className":12298,"style":871},[870],[59,12300,11054],{"className":12301,"style":11075},[103,104],[59,12303,6349],{"className":12304},[103],[14,12306,12307,12308,10201],{},"第一部分：当 ",[59,12309,12311,12345],{"className":12310,"translate":63},[62],[59,12312,12314],{"className":12313},[67],[69,12315,12316],{"xmlns":71},[73,12317,12318,12342],{},[76,12319,12320,12322,12328,12330,12332,12334,12336,12338,12340],{},[79,12321,6298],{"mathvariant":1091},[2924,12323,12324,12326],{},[79,12325,8477],{},[79,12327,2119],{},[79,12329,1092],{"mathvariant":1091},[79,12331,2119],{},[137,12333,6309],{},[79,12335,3432],{},[79,12337,6298],{"mathvariant":1091},[137,12339,6322],{},[79,12341,11054],{},[83,12343,12344],{"encoding":85},"|X_n\u002Fn-x|\u003C\\delta",[59,12346,12348,12412,12433],{"className":12347,"ariaHidden":90},[89],[59,12349,12351,12354,12357,12397,12400,12403,12406,12409],{"className":12350},[94],[59,12352],{"className":12353,"style":386},[98],[59,12355,6298],{"className":12356},[103],[59,12358,12360,12363],{"className":12359},[103],[59,12361,8477],{"className":12362,"style":8535},[103,104],[59,12364,12366],{"className":12365},[162],[59,12367,12369,12389],{"className":12368},[166,527],[59,12370,12372,12386],{"className":12371},[170],[59,12373,12375],{"className":12374,"style":6627},[174],[59,12376,12377,12380],{"style":8550},[59,12378],{"className":12379,"style":182},[181],[59,12381,12383],{"className":12382},[186,187,188,189],[59,12384,2119],{"className":12385},[103,104,189],[59,12387,564],{"className":12388},[563],[59,12390,12392],{"className":12391},[170],[59,12393,12395],{"className":12394,"style":3092},[174],[59,12396],{},[59,12398,1092],{"className":12399},[103],[59,12401,2119],{"className":12402},[103,104],[59,12404],{"className":12405,"style":105},[870],[59,12407,6309],{"className":12408},[193],[59,12410],{"className":12411,"style":105},[870],[59,12413,12415,12418,12421,12424,12427,12430],{"className":12414},[94],[59,12416],{"className":12417,"style":386},[98],[59,12419,3432],{"className":12420},[103,104],[59,12422,6298],{"className":12423},[103],[59,12425],{"className":12426,"style":871},[870],[59,12428,6322],{"className":12429},[875],[59,12431],{"className":12432,"style":871},[870],[59,12434,12436,12439],{"className":12435},[94],[59,12437],{"className":12438,"style":11179},[98],[59,12440,11054],{"className":12441,"style":11075},[103,104],[59,12443,12445],{"className":12444,"translate":63},[767],[59,12446,12448,12500],{"className":12447,"translate":63},[62],[59,12449,12451],{"className":12450},[67],[69,12452,12453],{"xmlns":71,"display":776},[73,12454,12455,12497],{},[76,12456,12457,12459,12461,12463,12469,12471,12473,12475,12477,12479,12481,12483,12485,12487,12489,12495],{},[79,12458,6298],{"mathvariant":1091},[79,12460,6010],{},[137,12462,368],{"stretchy":367},[2924,12464,12465,12467],{},[79,12466,8477],{},[79,12468,2119],{},[79,12470,1092],{"mathvariant":1091},[79,12472,2119],{},[137,12474,373],{"stretchy":367},[137,12476,6309],{},[79,12478,6010],{},[137,12480,368],{"stretchy":367},[79,12482,3432],{},[137,12484,373],{"stretchy":367},[79,12486,6298],{"mathvariant":1091},[137,12488,6322],{},[1174,12490,12491,12493],{},[79,12492,6192],{},[473,12494,475],{},[79,12496,6349],{"mathvariant":1091},[83,12498,12499],{"encoding":85},"|f(X_n\u002Fn)-f(x)|\n\u003C\n\\frac{\\varepsilon}{2}.",[59,12501,12503,12576,12606],{"className":12502,"ariaHidden":90},[89],[59,12504,12506,12509,12512,12515,12518,12558,12561,12564,12567,12570,12573],{"className":12505},[94],[59,12507],{"className":12508,"style":386},[98],[59,12510,6298],{"className":12511},[103],[59,12513,6010],{"className":12514,"style":6047},[103,104],[59,12516,368],{"className":12517},[393],[59,12519,12521,12524],{"className":12520},[103],[59,12522,8477],{"className":12523,"style":8535},[103,104],[59,12525,12527],{"className":12526},[162],[59,12528,12530,12550],{"className":12529},[166,527],[59,12531,12533,12547],{"className":12532},[170],[59,12534,12536],{"className":12535,"style":6627},[174],[59,12537,12538,12541],{"style":8550},[59,12539],{"className":12540,"style":182},[181],[59,12542,12544],{"className":12543},[186,187,188,189],[59,12545,2119],{"className":12546},[103,104,189],[59,12548,564],{"className":12549},[563],[59,12551,12553],{"className":12552},[170],[59,12554,12556],{"className":12555,"style":3092},[174],[59,12557],{},[59,12559,1092],{"className":12560},[103],[59,12562,2119],{"className":12563},[103,104],[59,12565,373],{"className":12566},[400],[59,12568],{"className":12569,"style":105},[870],[59,12571,6309],{"className":12572},[193],[59,12574],{"className":12575,"style":105},[870],[59,12577,12579,12582,12585,12588,12591,12594,12597,12600,12603],{"className":12578},[94],[59,12580],{"className":12581,"style":386},[98],[59,12583,6010],{"className":12584,"style":6047},[103,104],[59,12586,368],{"className":12587},[393],[59,12589,3432],{"className":12590},[103,104],[59,12592,373],{"className":12593},[400],[59,12595,6298],{"className":12596},[103],[59,12598],{"className":12599,"style":871},[870],[59,12601,6322],{"className":12602},[875],[59,12604],{"className":12605,"style":871},[870],[59,12607,12609,12612,12674],{"className":12608},[94],[59,12610],{"className":12611,"style":11305},[98],[59,12613,12615,12618,12671],{"className":12614},[103],[59,12616],{"className":12617},[393,1222],[59,12619,12621],{"className":12620},[1174],[59,12622,12624,12663],{"className":12623},[166,527],[59,12625,12627,12660],{"className":12626},[170],[59,12628,12630,12641,12649],{"className":12629,"style":6877},[174],[59,12631,12632,12635],{"style":4109},[59,12633],{"className":12634,"style":541},[181],[59,12636,12638],{"className":12637},[103],[59,12639,475],{"className":12640},[103],[59,12642,12643,12646],{"style":1253},[59,12644],{"className":12645,"style":541},[181],[59,12647],{"className":12648,"style":1261},[1260],[59,12650,12651,12654],{"style":4203},[59,12652],{"className":12653,"style":541},[181],[59,12655,12657],{"className":12656},[103],[59,12658,6192],{"className":12659},[103,104],[59,12661,564],{"className":12662},[563],[59,12664,12666],{"className":12665},[170],[59,12667,12669],{"className":12668,"style":4298},[174],[59,12670],{},[59,12672],{"className":12673},[400,1222],[59,12675,6349],{"className":12676},[103],[14,12678,12679,12680,47],{},"这部分贡献不超过 ",[59,12681,12683,12701],{"className":12682,"translate":63},[62],[59,12684,12686],{"className":12685},[67],[69,12687,12688],{"xmlns":71},[73,12689,12690,12698],{},[76,12691,12692],{},[1174,12693,12694,12696],{},[79,12695,6192],{},[473,12697,475],{},[83,12699,12700],{"encoding":85},"\\frac{\\varepsilon}{2}",[59,12702,12704],{"className":12703,"ariaHidden":90},[89],[59,12705,12707,12711],{"className":12706},[94],[59,12708],{"className":12709,"style":12710},[98],"height:1.0404em;vertical-align:-0.345em;",[59,12712,12714,12717,12777],{"className":12713},[103],[59,12715],{"className":12716},[393,1222],[59,12718,12720],{"className":12719},[1174],[59,12721,12723,12769],{"className":12722},[166,527],[59,12724,12726,12766],{"className":12725},[170],[59,12727,12730,12744,12752],{"className":12728,"style":12729},[174],"height:0.6954em;",[59,12731,12732,12735],{"style":1238},[59,12733],{"className":12734,"style":541},[181],[59,12736,12738],{"className":12737},[186,187,188,189],[59,12739,12741],{"className":12740},[103,189],[59,12742,475],{"className":12743},[103,189],[59,12745,12746,12749],{"style":1253},[59,12747],{"className":12748,"style":541},[181],[59,12750],{"className":12751,"style":1261},[1260],[59,12753,12754,12757],{"style":1264},[59,12755],{"className":12756,"style":541},[181],[59,12758,12760],{"className":12759},[186,187,188,189],[59,12761,12763],{"className":12762},[103,189],[59,12764,6192],{"className":12765},[103,104,189],[59,12767,564],{"className":12768},[563],[59,12770,12772],{"className":12771},[170],[59,12773,12775],{"className":12774,"style":1286},[174],[59,12776],{},[59,12778],{"className":12779},[400,1222],[14,12781,12782,12783,12930],{},"第二部分：设 ",[59,12784,12786,12831],{"className":12785,"translate":63},[62],[59,12787,12789],{"className":12788},[67],[69,12790,12791],{"xmlns":71},[73,12792,12793,12828],{},[76,12794,12795,12798,12800,12822,12824,12826],{},[79,12796,12797],{},"M",[137,12799,791],{},[2924,12801,12802,12810],{},[76,12803,12804,12807],{},[79,12805,12806],{},"max",[137,12808,12809],{},"⁡",[76,12811,12812,12814,12816,12818,12820],{},[137,12813,455],{"stretchy":367},[473,12815,471],{},[137,12817,1345],{"separator":90},[473,12819,637],{},[137,12821,487],{"stretchy":367},[79,12823,6298],{"mathvariant":1091},[79,12825,6010],{},[79,12827,6298],{"mathvariant":1091},[83,12829,12830],{"encoding":85},"M=\\max_{[0,1]}|f|",[59,12832,12834,12853],{"className":12833,"ariaHidden":90},[89],[59,12835,12837,12840,12844,12847,12850],{"className":12836},[94],[59,12838],{"className":12839,"style":99},[98],[59,12841,12797],{"className":12842,"style":12843},[103,104],"margin-right:0.109em;",[59,12845],{"className":12846,"style":871},[870],[59,12848,791],{"className":12849},[875],[59,12851],{"className":12852,"style":871},[870],[59,12854,12856,12860,12918,12921,12924,12927],{"className":12855},[94],[59,12857],{"className":12858,"style":12859},[98],"height:1.1052em;vertical-align:-0.3552em;",[59,12861,12863,12866],{"className":12862},[6688],[59,12864,12806],{"className":12865},[6688],[59,12867,12869],{"className":12868},[162],[59,12870,12872,12909],{"className":12871},[166,527],[59,12873,12875,12906],{"className":12874},[170],[59,12876,12879],{"className":12877,"style":12878},[174],"height:0.3448em;",[59,12880,12882,12885],{"style":12881},"top:-2.5198em;margin-right:0.05em;",[59,12883],{"className":12884,"style":182},[181],[59,12886,12888],{"className":12887},[186,187,188,189],[59,12889,12891,12894,12897,12900,12903],{"className":12890},[103,189],[59,12892,455],{"className":12893},[393,189],[59,12895,471],{"className":12896},[103,189],[59,12898,1345],{"className":12899},[1372,189],[59,12901,637],{"className":12902},[103,189],[59,12904,487],{"className":12905},[400,189],[59,12907,564],{"className":12908},[563],[59,12910,12912],{"className":12911},[170],[59,12913,12916],{"className":12914,"style":12915},[174],"height:0.3552em;",[59,12917],{},[59,12919],{"className":12920,"style":983},[870],[59,12922,6298],{"className":12923},[103],[59,12925,6010],{"className":12926,"style":6047},[103,104],[59,12928,6298],{"className":12929},[103],"。则",[59,12932,12934],{"className":12933,"translate":63},[767],[59,12935,12937,12987],{"className":12936,"translate":63},[62],[59,12938,12940],{"className":12939},[67],[69,12941,12942],{"xmlns":71,"display":776},[73,12943,12944,12984],{},[76,12945,12946,12948,12950,12952,12958,12960,12962,12964,12966,12968,12970,12972,12974,12976,12978,12980,12982],{},[79,12947,6298],{"mathvariant":1091},[79,12949,6010],{},[137,12951,368],{"stretchy":367},[2924,12953,12954,12956],{},[79,12955,8477],{},[79,12957,2119],{},[79,12959,1092],{"mathvariant":1091},[79,12961,2119],{},[137,12963,373],{"stretchy":367},[137,12965,6309],{},[79,12967,6010],{},[137,12969,368],{"stretchy":367},[79,12971,3432],{},[137,12973,373],{"stretchy":367},[79,12975,6298],{"mathvariant":1091},[137,12977,11787],{},[473,12979,475],{},[79,12981,12797],{},[79,12983,6349],{"mathvariant":1091},[83,12985,12986],{"encoding":85},"|f(X_n\u002Fn)-f(x)|\n\\le\n2M.",[59,12988,12990,13063,13093],{"className":12989,"ariaHidden":90},[89],[59,12991,12993,12996,12999,13002,13005,13045,13048,13051,13054,13057,13060],{"className":12992},[94],[59,12994],{"className":12995,"style":386},[98],[59,12997,6298],{"className":12998},[103],[59,13000,6010],{"className":13001,"style":6047},[103,104],[59,13003,368],{"className":13004},[393],[59,13006,13008,13011],{"className":13007},[103],[59,13009,8477],{"className":13010,"style":8535},[103,104],[59,13012,13014],{"className":13013},[162],[59,13015,13017,13037],{"className":13016},[166,527],[59,13018,13020,13034],{"className":13019},[170],[59,13021,13023],{"className":13022,"style":6627},[174],[59,13024,13025,13028],{"style":8550},[59,13026],{"className":13027,"style":182},[181],[59,13029,13031],{"className":13030},[186,187,188,189],[59,13032,2119],{"className":13033},[103,104,189],[59,13035,564],{"className":13036},[563],[59,13038,13040],{"className":13039},[170],[59,13041,13043],{"className":13042,"style":3092},[174],[59,13044],{},[59,13046,1092],{"className":13047},[103],[59,13049,2119],{"className":13050},[103,104],[59,13052,373],{"className":13053},[400],[59,13055],{"className":13056,"style":105},[870],[59,13058,6309],{"className":13059},[193],[59,13061],{"className":13062,"style":105},[870],[59,13064,13066,13069,13072,13075,13078,13081,13084,13087,13090],{"className":13065},[94],[59,13067],{"className":13068,"style":386},[98],[59,13070,6010],{"className":13071,"style":6047},[103,104],[59,13073,368],{"className":13074},[393],[59,13076,3432],{"className":13077},[103,104],[59,13079,373],{"className":13080},[400],[59,13082,6298],{"className":13083},[103],[59,1308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",[59,13112,13114,13160],{"className":13113,"translate":63},[62],[59,13115,13117],{"className":13116},[67],[69,13118,13119],{"xmlns":71},[73,13120,13121,13157],{},[76,13122,13123,13125,13127,13129,13131,13133,13135,13141,13143,13145,13147,13149,13151,13153,13155],{},[473,13124,475],{},[79,13126,12797],{},[137,13128,2316],{"separator":90},[79,13130,6249],{},[137,13132,368],{"stretchy":367},[79,13134,6298],{"mathvariant":1091},[2924,13136,13137,13139],{},[79,13138,8477],{},[79,13140,2119],{},[79,13142,1092],{"mathvariant":1091},[79,13144,2119],{},[137,13146,6309],{},[79,13148,3432],{},[79,13150,6298],{"mathvariant":1091},[137,13152,12162],{},[79,13154,11054],{},[137,13156,373],{"stretchy":367},[83,13158,13159],{"encoding":85},"2M·P(|X_n\u002Fn-x|\\ge\\delta)",[59,13161,13163,13245,13266],{"className":13162,"ariaHidden":90},[89],[59,13164,13166,13169,13172,13175,13178,13181,13184,13187,13190,13230,13233,13236,13239,13242],{"className":13165},[94],[59,13167],{"className":13168,"style":386},[98],[59,13170,475],{"className":13171},[103],[59,13173,12797],{"className":13174,"style":12843},[103,104],[59,13176,2316],{"className":13177},[1372],[59,13179],{"className":13180,"style":983},[870],[59,13182,6249],{"className":13183,"style":1412},[103,104],[59,13185,368],{"className":13186},[393],[59,13188,6298],{"className":13189},[103],[59,13191,13193,13196],{"className":13192},[103],[59,13194,8477],{"className":13195,"style":8535},[103,104],[59,13197,13199],{"className":13198},[162],[59,13200,13202,13222],{"className":13201},[166,527],[59,13203,13205,13219],{"className":13204},[170],[59,13206,13208],{"className":13207,"style":6627},[174],[59,13209,13210,13213],{"style":8550},[59,13211],{"className":13212,"style":182},[181],[59,13214,13216],{"className":13215},[186,187,188,189],[59,13217,2119],{"className":13218},[103,104,189],[59,13220,564],{"className":13221},[563],[59,13223,13225],{"className":13224},[170],[59,13226,13228],{"className":13227,"style":3092},[174],[59,13229],{},[59,13231,1092],{"className":13232},[103],[59,13234,2119],{"className":13235},[103,104],[59,13237],{"className":13238,"style":105},[870],[59,13240,6309],{"className":13241},[193],[59,13243],{"className":13244,"style":105},[870],[59,13246,13248,13251,13254,13257,13260,13263],{"className":13247},[94],[59,13249],{"className":13250,"style":386},[98],[59,13252,3432],{"className":13253},[103,104],[59,13255,6298],{"className":13256},[103],[59,13258],{"className":13259,"style":871},[870],[59,13261,12162],{"className":13262},[875],[59,13264],{"className":13265,"style":871},[870],[59,13267,13269,13272,13275],{"className":13268},[94],[59,13270],{"className":13271,"style":386},[98],[59,13273,11054],{"className":13274,"style":11075},[103,104],[59,13276,373],{"className":13277},[400],[14,13279,13280],{},"利用切比雪夫不等式：",[59,13282,13284],{"className":13283,"translate":63},[767],[59,13285,13287,13375],{"className":13286,"translate":63},[62],[59,13288,13290],{"className":13289},[67],[69,13291,13292],{"xmlns":71,"display":776},[73,13293,13294,13372],{},[76,13295,13296,13298,13300,13302,13308,13310,13312,13314,13316,13318,13320,13322,13324,13326,13352,13354,13370],{},[79,13297,6249],{},[137,13299,368],{"stretchy":367},[79,13301,6298],{"mathvariant":1091},[2924,13303,13304,13306],{},[79,13305,8477],{},[79,13307,2119],{},[79,13309,1092],{"mathvariant":1091},[79,13311,2119],{},[137,13313,6309],{},[79,13315,3432],{},[79,13317,6298],{"mathvariant":1091},[137,13319,12162],{},[79,13321,11054],{},[137,13323,373],{"stretchy":367},[137,13325,11787],{},[1174,13327,13328,13342],{},[76,13329,13330,13332,13334,13336,13338,13340],{},[79,13331,3432],{},[137,13333,368],{"stretchy":367},[473,13335,637],{},[137,13337,6309],{},[79,13339,3432],{},[137,13341,373],{"stretchy":367},[76,13343,13344,13346],{},[79,13345,2119],{},[132,13347,13348,13350],{},[79,13349,11054],{},[473,13351,475],{},[137,13353,11787],{},[1174,13355,13356,13358],{},[473,13357,637],{},[76,13359,13360,13362,13364],{},[473,13361,644],{},[79,13363,2119],{},[132,13365,13366,13368],{},[79,13367,11054],{},[473,13369,475],{},[79,13371,6349],{"mathvariant":1091},[83,13373,13374],{"encoding":85},"P(|X_n\u002Fn-x|\\ge\\delta)\n\\le\n\\frac{x(1-x)}\n{n\\delta^2}\n\\le\n\\frac1{4n\\delta^2}.",[59,13376,13378,13448,13469,13490,13618],{"className":13377,"ariaHidden":90},[89],[59,13379,13381,13384,13387,13390,13393,13433,13436,13439,13442,13445],{"className":13380},[94],[59,13382],{"className":13383,"style":386},[98],[59,13385,6249],{"className":13386,"style":1412},[103,104],[59,13388,368],{"className":13389},[393],[59,13391,6298],{"className":13392},[103],[59,13394,13396,13399],{"className":13395},[103],[59,13397,8477],{"className":13398,"style":8535},[103,104],[59,13400,13402],{"className":13401},[162],[59,13403,13405,13425],{"className":13404},[166,527],[59,13406,13408,13422],{"className":13407},[170],[59,13409,13411],{"className":13410,"style":6627},[174],[59,13412,13413,13416],{"style":8550},[59,13414],{"className":13415,"style":182},[181],[59,13417,13419],{"className":13418},[186,187,188,189],[59,13420,2119],{"className":13421},[103,104,189],[59,13423,564],{"className":13424},[563],[59,13426,13428],{"className":13427},[170],[59,13429,13431],{"className":13430,"style":3092},[174],[59,13432],{},[59,13434,1092],{"className":13435},[103],[59,13437,2119],{"className":13438},[103,104],[59,13440],{"className":13441,"style":105},[870],[59,13443,6309],{"className":13444},[193],[59,13446],{"className":13447,"style":105},[870],[59,13449,13451,13454,13457,13460,13463,13466],{"className":13450},[94],[59,13452],{"className":13453,"style":386},[98],[59,13455,3432],{"className":13456},[103,104],[59,13458,6298],{"className":13459},[103],[59,13461],{"className":13462,"style":871},[870],[59,13464,12162],{"className":13465},[875],[59,13467],{"className":13468,"style":871},[870],[59,13470,13472,13475,13478,13481,13484,13487],{"className":13471},[94],[59,13473],{"className":13474,"style":386},[98],[59,13476,11054],{"className":13477,"style":11075},[103,104],[59,13479,373],{"className":13480},[400],[59,13482],{"className":13483,"style":871},[870],[59,13485,11787],{"className":13486},[875],[59,13488],{"className":13489,"style":871},[870],[59,13491,13493,13496,13609,13612,13615],{"className":13492},[94],[59,13494],{"className":13495,"style":9886},[98],[59,13497,13499,13502,13606],{"className":13498},[103],[59,13500],{"className":13501},[393,1222],[59,13503,13505],{"className":13504},[1174],[59,13506,13508,13598],{"className":13507},[166,527],[59,13509,13511,13595],{"className":13510},[170],[59,13512,13514,13555,13563],{"className":13513,"style":9905},[174],[59,13515,13516,13519],{"style":4109},[59,13517],{"className":13518,"style":541},[181],[59,13520,13522,13525],{"className":13521},[103],[59,13523,2119],{"className":13524},[103,104],[59,13526,13528,13531],{"className":13527},[103],[59,13529,11054],{"className":13530,"style":11075},[103,104],[59,13532,13534],{"className":13533},[162],[59,13535,13537],{"className":13536},[166],[59,13538,13540],{"className":13539},[170],[59,13541,13544],{"className":13542,"style":13543},[174],"height:0.7401em;",[59,13545,13546,13549],{"style":4141},[59,13547],{"className":13548,"style":182},[181],[59,13550,13552],{"className":13551},[186,187,188,189],[59,13553,475],{"className":13554},[103,189],[59,13556,13557,13560],{"style":1253},[59,13558],{"className":13559,"style":541},[181],[59,13561],{"className":13562,"style":1261},[1260],[59,13564,13565,13568],{"style":4203},[59,13566],{"className":13567,"style":541},[181],[59,13569,13571,13574,13577,13580,13583,13586,13589,13592],{"className":13570},[103],[59,13572,3432],{"className":13573},[103,104],[59,13575,368],{"className":13576},[393],[59,13578,637],{"className":13579},[103],[59,13581],{"className":13582,"style":105},[870],[59,13584,6309],{"className":13585},[193],[59,13587],{"className":13588,"style":105},[870],[59,13590,3432],{"className":13591},[103,104],[59,13593,373],{"className":13594},[400],[59,13596,564],{"className":13597},[563],[59,13599,13601],{"className":13600},[170],[59,13602,13604],{"className":13603,"style":4298},[174],[59,13605],{},[59,13607],{"className":13608},[400,1222],[59,13610],{"className":13611,"style":871},[870],[59,13613,11787],{"className":13614},[875],[59,13616],{"className":13617,"style":871},[870],[59,13619,13621,13625,13719],{"className":13620},[94],[59,13622],{"className":13623,"style":13624},[98],"height:2.0074em;vertical-align:-0.686em;",[59,13626,13628,13631,13716],{"className":13627},[103],[59,13629],{"className":13630},[393,1222],[59,13632,13634],{"className":13633},[1174],[59,13635,13637,13708],{"className":13636},[166,527],[59,13638,13640,13705],{"className":13639},[170],[59,13641,13643,13686,13694],{"className":13642,"style":8045},[174],[59,13644,13645,13648],{"style":4109},[59,13646],{"className":13647,"style":541},[181],[59,13649,13651,13654,13657],{"className":13650},[103],[59,13652,644],{"className":13653},[103],[59,13655,2119],{"className":13656},[103,104],[59,13658,13660,13663],{"className":13659},[103],[59,13661,11054],{"className":13662,"style":11075},[103,104],[59,13664,13666],{"className":13665},[162],[59,13667,13669],{"className":13668},[166],[59,13670,13672],{"className":13671},[170],[59,13673,13675],{"className":13674,"style":13543},[174],[59,13676,13677,13680],{"style":4141},[59,13678],{"className":13679,"style":182},[181],[59,13681,13683],{"className":13682},[186,187,188,189],[59,13684,475],{"className":13685},[103,189],[59,13687,13688,13691],{"style":1253},[59,13689],{"className":13690,"style":541},[181],[59,13692],{"className":13693,"style":1261},[1260],[59,13695,13696,13699],{"style":4203},[59,13697],{"className":13698,"style":541},[181],[59,13700,13702],{"className":13701},[103],[59,13703,637],{"className":13704},[103],[59,13706,564],{"className":13707},[563],[59,13709,13711],{"className":13710},[170],[59,13712,13714],{"className":13713,"style":4298},[174],[59,13715],{},[59,13717],{"className":13718},[400,1222],[59,13720,6349],{"className":13721},[103],[14,13723,7269],{},[59,13725,13727],{"className":13726,"translate":63},[767],[59,13728,13730,13796],{"className":13729,"translate":63},[62],[59,13731,13733],{"className":13732},[67],[69,13734,13735],{"xmlns":71,"display":776},[73,13736,13737,13793],{},[76,13738,13739,13741,13743,13745,13747,13749,13751,13757,13759,13761,13763,13765,13767,13769,13771,13773,13775,13791],{},[473,13740,475],{},[79,13742,12797],{},[137,13744,2316],{"separator":90},[79,13746,6249],{},[137,13748,368],{"stretchy":367},[79,13750,6298],{"mathvariant":1091},[2924,13752,13753,13755],{},[79,13754,8477],{},[79,13756,2119],{},[79,13758,1092],{"mathvariant":1091},[79,13760,2119],{},[137,13762,6309],{},[79,13764,3432],{},[79,13766,6298],{"mathvariant":1091},[137,13768,12162],{},[79,13770,11054],{},[137,13772,373],{"stretchy":367},[137,13774,11787],{},[1174,13776,13777,13779],{},[79,13778,12797],{},[76,13780,13781,13783,13785],{},[473,13782,475],{},[79,13784,2119],{},[132,13786,13787,13789],{},[79,13788,11054],{},[473,13790,475],{},[79,13792,6349],{"mathvariant":1091},[83,13794,13795],{"encoding":85},"2M·\nP(|X_n\u002Fn-x|\\ge\\delta)\n\\le\n\\frac{M}{2n\\delta^2}.",[59,13797,13799,13881,13902,13923],{"className":13798,"ariaHidden":90},[89],[59,13800,13802,13805,13808,13811,13814,13817,13820,13823,13826,13866,13869,13872,13875,13878],{"className":13801},[94],[59,13803],{"className":13804,"style":386},[98],[59,13806,475],{"className":13807},[103],[59,13809,12797],{"className":13810,"style":12843},[103,104],[59,13812,2316],{"className":13813},[1372],[59,13815],{"className":13816,"style":983},[870],[59,13818,6249],{"className":13819,"style":1412},[103,104],[59,13821,368],{"className":13822},[393],[59,13824,6298],{"className":13825},[103],[59,13827,13829,13832],{"className":13828},[103],[59,13830,8477],{"className":13831,"style":8535},[103,104],[59,13833,13835],{"className":13834},[162],[59,13836,13838,13858],{"className":13837},[166,527],[59,13839,13841,13855],{"className":13840},[170],[59,13842,13844],{"className":13843,"style":6627},[174],[59,13845,13846,13849],{"style":8550},[59,13847],{"className":13848,"style":182},[181],[59,13850,13852],{"className":13851},[186,187,188,189],[59,13853,2119],{"className":13854},[103,104,189],[59,13856,564],{"className":13857},[563],[59,13859,13861],{"className":13860},[170],[59,13862,13864],{"className":13863,"style":3092},[174],[59,13865],{},[59,13867,1092],{"className":13868},[103],[59,13870,2119],{"className":13871},[103,104],[59,13873],{"className":13874,"style":105},[870],[59,13876,6309],{"className":13877},[193],[59,13879],{"className":13880,"style":105},[870],[59,13882,13884,13887,13890,13893,13896,13899],{"className":13883},[94],[59,13885],{"className":13886,"style":386},[98],[59,13888,3432],{"className":13889},[103,104],[59,13891,6298],{"className":13892},[103],[59,13894],{"className":13895,"style":871},[870],[59,13897,12162],{"className":13898},[875],[59,13900],{"className":13901,"style":871},[870],[59,13903,13905,13908,13911,13914,13917,13920],{"className":13904},[94],[59,13906],{"className":13907,"style":386},[98],[59,13909,11054],{"className":13910,"style":11075},[103,104],[59,13912,373],{"className":13913},[400],[59,13915],{"className":13916,"style":871},[870],[59,13918,11787],{"className":13919},[875],[59,13921],{"className":13922,"style":871},[870],[59,13924,13926,13929,14023],{"className":13925},[94],[59,13927],{"className":13928,"style":10013},[98],[59,13930,13932,13935,14020],{"className":13931},[103],[59,13933],{"className":13934},[393,1222],[59,13936,13938],{"className":13937},[1174],[59,13939,13941,14012],{"className":13940},[166,527],[59,13942,13944,14009],{"className":13943},[170],[59,13945,13947,13990,13998],{"className":13946,"style":9150},[174],[59,13948,13949,13952],{"style":4109},[59,13950],{"className":13951,"style":541},[181],[59,13953,13955,13958,13961],{"className":13954},[103],[59,13956,475],{"className":13957},[103],[59,13959,2119],{"className":13960},[103,104],[59,13962,13964,13967],{"className":13963},[103],[59,13965,11054],{"className":13966,"style":11075},[103,104],[59,13968,13970],{"className":13969},[162],[59,13971,13973],{"className":13972},[166],[59,13974,13976],{"className":13975},[170],[59,13977,13979],{"className":13978,"style":13543},[174],[59,13980,13981,13984],{"style":4141},[59,13982],{"className":13983,"style":182},[181],[59,13985,13987],{"className":13986},[186,187,188,189],[59,13988,475],{"className":13989},[103,189],[59,13991,13992,13995],{"style":1253},[59,13993],{"className":13994,"style":541},[181],[59,13996],{"className":13997,"style":1261},[1260],[59,13999,14000,14003],{"style":4203},[59,14001],{"className":14002,"style":541},[181],[59,14004,14006],{"className":14005},[103],[59,14007,12797],{"className":14008,"style":12843},[103,104],[59,14010,564],{"className":14011},[563],[59,14013,14015],{"className":14014},[170],[59,14016,14018],{"className":14017,"style":4298},[174],[59,14019],{},[59,14021],{"className":14022},[400,1222],[59,14024,6349],{"className":14025},[103],[14,14027,10147,14028,14056],{},[59,14029,14031,14044],{"className":14030,"translate":63},[62],[59,14032,14034],{"className":14033},[67],[69,14035,14036],{"xmlns":71},[73,14037,14038,14042],{},[76,14039,14040],{},[79,14041,2119],{},[83,14043,2119],{"encoding":85},[59,14045,14047],{"className":14046,"ariaHidden":90},[89],[59,14048,14050,14053],{"className":14049},[94],[59,14051],{"className":14052,"style":310},[98],[59,14054,2119],{"className":14055},[103,104]," 足够大时，",[59,14058,14060],{"className":14059,"translate":63},[767],[59,14061,14063,14101],{"className":14062,"translate":63},[62],[59,14064,14066],{"className":14065},[67],[69,14067,14068],{"xmlns":71,"display":776},[73,14069,14070,14098],{},[76,14071,14072,14088,14090,14096],{},[1174,14073,14074,14076],{},[79,14075,12797],{},[76,14077,14078,14080,14082],{},[473,14079,475],{},[79,14081,2119],{},[132,14083,14084,14086],{},[79,14085,11054],{},[473,14087,475],{},[137,14089,6322],{},[1174,14091,14092,14094],{},[79,14093,6192],{},[473,14095,475],{},[79,14097,6349],{"mathvariant":1091},[83,14099,14100],{"encoding":85},"\\frac{M}{2n\\delta^2}\n\u003C\n\\frac{\\varepsilon}{2}.",[59,14102,14104,14213],{"className":14103,"ariaHidden":90},[89],[59,14105,14107,14110,14204,14207,14210],{"className":14106},[94],[59,14108],{"className":14109,"style":10013},[98],[59,14111,14113,14116,14201],{"className":14112},[103],[59,14114],{"className":14115},[393,1222],[59,14117,14119],{"className":14118},[1174],[59,14120,14122,14193],{"className":14121},[166,527],[59,14123,14125,14190],{"className":14124},[170],[59,14126,14128,14171,14179],{"className":14127,"style":9150},[174],[59,14129,14130,14133],{"style":4109},[59,14131],{"className":14132,"style":541},[181],[59,14134,14136,14139,14142],{"className":14135},[103],[59,14137,475],{"className":14138},[103],[59,14140,2119],{"className":14141},[103,104],[59,14143,14145,14148],{"className":14144},[103],[59,14146,11054],{"className":14147,"style":11075},[103,104],[59,14149,14151],{"className":14150},[162],[59,14152,14154],{"className":14153},[166],[59,14155,14157],{"className":14156},[170],[59,14158,14160],{"className":14159,"style":13543},[174],[59,14161,14162,14165],{"style":4141},[59,14163],{"className":14164,"style":182},[181],[59,14166,14168],{"className":14167},[186,187,188,189],[59,14169,475],{"className":14170},[103,189],[59,14172,14173,14176],{"style":1253},[59,14174],{"className":14175,"style":541},[181],[59,14177],{"className":14178,"style":1261},[1260],[59,14180,14181,14184],{"style":4203},[59,14182],{"className":14183,"style":541},[181],[59,14185,14187],{"className":14186},[103],[59,14188,12797],{"className":14189,"style":12843},[103,104],[59,14191,564],{"className":14192},[563],[59,14194,14196],{"className":14195},[170],[59,14197,14199],{"className":14198,"style":4298},[174],[59,14200],{},[59,14202],{"className":14203},[400,1222],[59,14205],{"className":14206,"style":871},[870],[59,14208,6322],{"className":14209},[875],[59,14211],{"className":14212,"style":871},[870],[59,14214,14216,14219,14281],{"className":14215},[94],[59,14217],{"className":14218,"style":11305},[98],[59,14220,14222,14225,14278],{"className":14221},[103],[59,14223],{"className":14224},[393,1222],[59,14226,14228],{"className":14227},[1174],[59,14229,14231,14270],{"className":14230},[166,527],[59,14232,14234,14267],{"className":14233},[170],[59,14235,14237,14248,14256],{"className":14236,"style":6877},[174],[59,14238,14239,14242],{"style":4109},[59,14240],{"className":14241,"style":541},[181],[59,14243,14245],{"className":14244},[103],[59,14246,475],{"className":14247},[103],[59,14249,14250,14253],{"style":1253},[59,14251],{"className":14252,"style":541},[181],[59,14254],{"className":14255,"style":1261},[1260],[59,14257,14258,14261],{"style":4203},[59,14259],{"className":14260,"style":541},[181],[59,14262,14264],{"className":14263},[103],[59,14265,6192],{"className":14266},[103,104],[59,14268,564],{"className":14269},[563],[59,14271,14273],{"className":14272},[170],[59,14274,14276],{"className":14275,"style":4298},[174],[59,14277],{},[59,14279],{"className":14280},[400,1222],[59,14282,6349],{"className":14283},[103],[14,14285,10367],{},[59,14287,14289],{"className":14288,"translate":63},[767],[59,14290,14292,14342],{"className":14291,"translate":63},[62],[59,14293,14295],{"className":14294},[67],[69,14296,14297],{"xmlns":71,"display":776},[73,14298,14299,14339],{},[76,14300,14301,14303,14309,14311,14313,14315,14317,14319,14321,14323,14325,14327,14329,14331,14333,14335,14337],{},[79,14302,6298],{"mathvariant":1091},[2924,14304,14305,14307],{},[79,14306,6505],{},[79,14308,2119],{},[137,14310,368],{"stretchy":367},[79,14312,6010],{},[137,14314,373],{"stretchy":367},[137,14316,368],{"stretchy":367},[79,14318,3432],{},[137,14320,373],{"stretchy":367},[137,14322,6309],{},[79,14324,6010],{},[137,14326,368],{"stretchy":367},[79,14328,3432],{},[137,14330,373],{"stretchy":367},[79,14332,6298],{"mathvariant":1091},[137,14334,6322],{},[79,14336,6192],{},[79,14338,6349],{"mathvariant":1091},[83,14340,14341],{"encoding":85},"|B_n(f)(x)-f(x)|\n\u003C\n\\varepsilon.",[59,14343,14345,14421,14451],{"className":14344,"ariaHidden":90},[89],[59,14346,14348,14351,14354,14394,14397,14400,14403,14406,14409,14412,14415,14418],{"className":14347},[94],[59,14349],{"className":14350,"style":386},[98],[59,14352,6298],{"className":14353},[103],[59,14355,14357,14360],{"className":14356},[103],[59,14358,6505],{"className":14359,"style":6614},[103,104],[59,14361,14363],{"className":14362},[162],[59,14364,14366,14386],{"className":14365},[166,527],[59,14367,14369,14383],{"className":14368},[170],[59,14370,14372],{"className":14371,"style":6627},[174],[59,14373,14374,14377],{"style":6630},[59,14375],{"className":14376,"style":182},[181],[59,14378,14380],{"className":14379},[186,187,188,189],[59,14381,2119],{"className":14382},[103,104,189],[59,14384,564],{"className":14385},[563],[59,14387,14389],{"className":14388},[170],[59,14390,14392],{"className":14391,"style":3092},[174],[59,14393],{},[59,14395,368],{"className":14396},[393],[59,14398,6010],{"className":14399,"style":6047},[103,104],[59,14401,373],{"className":14402},[400],[59,14404,368],{"className":14405},[393],[59,14407,3432],{"className":14408},[103,104],[59,14410,373],{"className":14411},[400],[59,14413],{"className":14414,"style":105},[870],[59,14416,6309],{"className":14417},[193],[59,14419],{"className":14420,"style":105},[870],[59,14422,14424,14427,14430,14433,14436,14439,14442,14445,14448],{"className":14423},[94],[59,14425],{"className":14426,"style":386},[98],[59,14428,6010],{"className":14429,"style":6047},[103,104],[59,14431,368],{"className":14432},[393],[59,14434,3432],{"className":14435},[103,104],[59,14437,373],{"className":14438},[400],[59,14440,6298],{"className":14441},[103],[59,14443],{"className":14444,"style":871},[870],[59,14446,6322],{"className":14447},[875],[59,14449],{"className":14450,"style":871},[870],[59,14452,14454,14457,14460],{"className":14453},[94],[59,14455],{"className":14456,"style":310},[98],[59,14458,6192],{"className":14459},[103,104],[59,14461,6349],{"className":14462},[103],[14,14464,14465,14466,14494],{},"而且这个估计与 ",[59,14467,14469,14482],{"className":14468,"translate":63},[62],[59,14470,14472],{"className":14471},[67],[69,14473,14474],{"xmlns":71},[73,14475,14476,14480],{},[76,14477,14478],{},[79,14479,3432],{},[83,14481,3432],{"encoding":85},[59,14483,14485],{"className":14484,"ariaHidden":90},[89],[59,14486,14488,14491],{"className":14487},[94],[59,14489],{"className":14490,"style":310},[98],[59,14492,3432],{"className":14493},[103,104]," 无关。",[14,14496,14497],{},"因此，",[59,14499,14501],{"className":14500,"translate":63},[767],[59,14502,14504,14578],{"className":14503,"translate":63},[62],[59,14505,14507],{"className":14506},[67],[69,14508,14509],{"xmlns":71,"display":776},[73,14510,14511,14575],{},[76,14512,14513,14538,14540,14546,14548,14550,14552,14554,14556,14558,14560,14562,14564,14566,14568,14570,14572],{},[1497,14514,14515,14522],{},[76,14516,14517,14520],{},[79,14518,14519],{},"sup",[137,14521,12809],{},[76,14523,14524,14526,14528,14530,14532,14534,14536],{},[79,14525,3432],{},[137,14527,6013],{},[137,14529,455],{"stretchy":367},[473,14531,471],{},[137,14533,1345],{"separator":90},[473,14535,637],{},[137,14537,487],{"stretchy":367},[79,14539,6298],{"mathvariant":1091},[2924,14541,14542,14544],{},[79,14543,6505],{},[79,14545,2119],{},[137,14547,368],{"stretchy":367},[79,14549,6010],{},[137,14551,373],{"stretchy":367},[137,14553,368],{"stretchy":367},[79,14555,3432],{},[137,14557,373],{"stretchy":367},[137,14559,6309],{},[79,14561,6010],{},[137,14563,368],{"stretchy":367},[79,14565,3432],{},[137,14567,373],{"stretchy":367},[79,14569,6298],{"mathvariant":1091},[137,14571,10164],{},[473,14573,14574],{},"0.",[83,14576,14577],{"encoding":85},"\\sup_{x\\in[0,1]}\n|B_n(f)(x)-f(x)|\n\\to 0.",[59,14579,14581,14728,14758],{"className":14580,"ariaHidden":90},[89],[59,14582,14584,14588,14655,14658,14661,14701,14704,14707,14710,14713,14716,14719,14722,14725],{"className":14583},[94],[59,14585],{"className":14586,"style":14587},[98],"height:1.9104em;vertical-align:-1.1604em;",[59,14589,14591],{"className":14590},[6688,6689],[59,14592,14594,14646],{"className":14593},[166,527],[59,14595,14597,14643],{"className":14596},[170],[59,14598,14600,14633],{"className":14599,"style":310},[174],[59,14601,14603,14606],{"style":14602},"top:-2.1146em;margin-left:0em;",[59,14604],{"className":14605,"style":541},[181],[59,14607,14609],{"className":14608},[186,187,188,189],[59,14610,14612,14615,14618,14621,14624,14627,14630],{"className":14611},[103,189],[59,14613,3432],{"className":14614},[103,104,189],[59,14616,6013],{"className":14617},[875,189],[59,14619,455],{"className":14620},[393,189],[59,14622,471],{"className":14623},[103,189],[59,14625,1345],{"className":14626},[1372,189],[59,14628,637],{"className":14629},[103,189],[59,14631,487],{"className":14632},[400,189],[59,14634,14635,14638],{"style":1617},[59,14636],{"className":14637,"style":541},[181],[59,14639,14640],{},[59,14641,14519],{"className":14642},[6688],[59,14644,564],{"className":14645},[563],[59,14647,14649],{"className":14648},[170],[59,14650,14653],{"className":14651,"style":14652},[174],"height:1.1604em;",[59,14654],{},[59,14656],{"className":14657,"style":983},[870],[59,14659,6298],{"className":14660},[103],[59,14662,14664,14667],{"className":14663},[103],[59,14665,6505],{"className":14666,"style":6614},[103,104],[59,14668,14670],{"className":14669},[162],[59,14671,14673,14693],{"className":14672},[166,527],[59,14674,14676,14690],{"className":14675},[170],[59,14677,14679],{"className":14678,"style":6627},[174],[59,14680,14681,14684],{"style":6630},[59,14682],{"className":14683,"style":182},[181],[59,14685,14687],{"className":14686},[186,187,188,189],[59,14688,2119],{"className":14689},[103,104,189],[59,14691,564],{"className":14692},[563],[59,14694,14696],{"className":14695},[170],[59,14697,14699],{"className":14698,"style":3092},[174],[59,14700],{},[59,14702,368],{"className":14703},[393],[59,14705,6010],{"className":14706,"style":6047},[103,104],[59,14708,373],{"className":14709},[400],[59,14711,368],{"className":14712},[393],[59,14714,3432],{"className":14715},[103,104],[59,14717,373],{"className":14718},[400],[59,14720],{"className":14721,"style":105},[870],[59,14723,6309],{"className":14724},[193],[59,14726],{"className":14727,"style":105},[870],[59,14729,14731,14734,14737,14740,14743,14746,14749,14752,14755],{"className":14730},[94],[59,14732],{"className":14733,"style":386},[98],[59,14735,6010],{"className":14736,"style":6047},[103,104],[59,14738,368],{"className":14739},[393],[59,14741,3432],{"className":14742},[103,104],[59,14744,373],{"className":14745},[400],[59,14747,6298],{"className":14748},[103],[59,14750],{"className":14751,"style":871},[870],[59,14753,10164],{"className":14754},[875],[59,14756],{"className":14757,"style":871},[870],[59,14759,14761,14764],{"className":14760},[94],[59,14762],{"className":14763,"style":1065},[98],[59,14765,14574],{"className":14766},[103],[14,14768,8617],{},[59,14770,14772],{"className":14771,"translate":63},[767],[59,14773,14775,14803],{"className":14774,"translate":63},[62],[59,14776,14778],{"className":14777},[67],[69,14779,14780],{"xmlns":71,"display":776},[73,14781,14782,14800],{},[76,14783,14784,14790,14792,14794,14796,14798],{},[2924,14785,14786,14788],{},[79,14787,6505],{},[79,14789,2119],{},[137,14791,368],{"stretchy":367},[79,14793,6010],{},[137,14795,373],{"stretchy":367},[137,14797,10164],{},[79,14799,6010],{},[83,14801,14802],{"encoding":85},"B_n(f)\n\\to f",[59,14804,14806,14870],{"className":14805,"ariaHidden":90},[89],[59,14807,14809,14812,14852,14855,14858,14861,14864,14867],{"className":14808},[94],[59,14810],{"className":14811,"style":386},[98],[59,14813,14815,14818],{"className":14814},[103],[59,14816,6505],{"className":14817,"style":6614},[103,104],[59,14819,14821],{"className":14820},[162],[59,14822,14824,14844],{"className":14823},[166,527],[59,14825,14827,14841],{"className":14826},[170],[59,14828,14830],{"className":14829,"style":6627},[174],[59,14831,14832,14835],{"style":6630},[59,14833],{"className":14834,"style":182},[181],[59,14836,14838],{"className":14837},[186,187,188,189],[59,14839,2119],{"className":14840},[103,104,189],[59,14842,564],{"className":14843},[563],[59,14845,14847],{"className":14846},[170],[59,14848,14850],{"className":14849,"style":3092},[174],[59,14851],{},[59,14853,368],{"className":14854},[393],[59,14856,6010],{"className":14857,"style":6047},[103,104],[59,14859,373],{"className":14860},[400],[59,14862],{"className":14863,"style":871},[870],[59,14865,10164],{"className":14866},[875],[59,14868],{"className":14869,"style":871},[870],[59,14871,14873,14876],{"className":14872},[94],[59,14874],{"className":14875,"style":6043},[98],[59,14877,6010],{"className":14878,"style":6047},[103,104],[14,14880,14881],{},"一致收敛。",[14,14883,14884],{},"魏尔斯特拉斯定理证毕。",[30,14886,14888],{"id":14887},"uat-的证明","UAT 的证明",[14,14890,14891],{},"接下来，我们用只含基本微积分就能理解的语言，逐步走一遍用多项式逼近证明通用逼近定理（UAT）的过程。为清晰起见，我们以一元连续函数为例；多元情形可类似推广。",[14,14893,14894,14895,14992,14993,15045],{},"目标：给定连续函数 ",[59,14896,14898,14929],{"className":14897,"translate":63},[62],[59,14899,14901],{"className":14900},[67],[69,14902,14903],{"xmlns":71},[73,14904,14905,14926],{},[76,14906,14907,14909,14911,14913,14915,14917,14920,14922,14924],{},[79,14908,6010],{},[137,14910,1495],{},[137,14912,455],{"stretchy":367},[79,14914,8501],{},[137,14916,1345],{"separator":90},[79,14918,14919],{},"b",[137,14921,487],{"stretchy":367},[137,14923,10164],{},[79,14925,1555],{"mathvariant":1554},[83,14927,14928],{"encoding":85},"f:[a,b]\\to \\mathbb{R}",[59,14930,14932,14950,14983],{"className":14931,"ariaHidden":90},[89],[59,14933,14935,14938,14941,14944,14947],{"className":14934},[94],[59,14936],{"className":14937,"style":6043},[98],[59,14939,6010],{"className":14940,"style":6047},[103,104],[59,14942],{"className":14943,"style":871},[870],[59,14945,1495],{"className":14946},[875],[59,14948],{"className":14949,"style":871},[870],[59,14951,14953,14956,14959,14962,14965,14968,14971,14974,14977,14980],{"className":14952},[94],[59,14954],{"className":14955,"style":386},[98],[59,14957,455],{"className":14958},[393],[59,14960,8501],{"className":14961},[103,104],[59,14963,1345],{"className":14964},[1372],[59,14966],{"className":14967,"style":983},[870],[59,14969,14919],{"className":14970},[103,104],[59,14972,487],{"className":14973},[400],[59,14975],{"className":14976,"style":871},[870],[59,14978,10164],{"className":14979},[875],[59,14981],{"className":14982,"style":871},[870],[59,14984,14986,14989],{"className":14985},[94],[59,14987],{"className":14988,"style":1956},[98],[59,14990,1555],{"className":14991},[103,1960]," 和任意 ",[59,14994,14996,15015],{"className":14995,"translate":63},[62],[59,14997,14999],{"className":14998},[67],[69,15000,15001],{"xmlns":71},[73,15002,15003,15012],{},[76,15004,15005,15008,15010],{},[79,15006,15007],{},"ϵ",[137,15009,6195],{},[473,15011,471],{},[83,15013,15014],{"encoding":85},"\\epsilon>0",[59,15016,15018,15036],{"className":15017,"ariaHidden":90},[89],[59,15019,15021,15024,15027,15030,15033],{"className":15020},[94],[59,15022],{"className":15023,"style":6210},[98],[59,15025,15007],{"className":15026},[103,104],[59,15028],{"className":15029,"style":871},[870],[59,15031,6195],{"className":15032},[875],[59,15034],{"className":15035,"style":871},[870],[59,15037,15039,15042],{"className":15038},[94],[59,15040],{"className":15041,"style":1065},[98],[59,15043,471],{"className":15044},[103],"，存在一个单隐层神经网络",[59,15047,15049],{"className":15048,"translate":63},[767],[59,15050,15052,15120],{"className":15051,"translate":63},[62],[59,15053,15055],{"className":15054},[67],[69,15056,15057],{"xmlns":71,"display":776},[73,15058,15059,15117],{},[76,15060,15061,15064,15066,15068,15070,15072,15086,15093,15095,15097,15104,15106,15108,15115],{},[79,15062,15063],{},"N",[137,15065,368],{"stretchy":367},[79,15067,3432],{},[137,15069,373],{"stretchy":367},[137,15071,791],{},[6523,15073,15074,15076,15084],{},[137,15075,6527],{},[76,15077,15078,15080,15082],{},[79,15079,2933],{},[137,15081,791],{},[473,15083,637],{},[79,15085,2119],{},[2924,15087,15088,15091],{},[79,15089,15090],{},"α",[79,15092,2933],{},[79,15094,2088],{},[137,15096,368],{"stretchy":367},[2924,15098,15099,15102],{},[79,15100,15101],{},"β",[79,15103,2933],{},[79,15105,3432],{},[137,15107,2000],{},[2924,15109,15110,15113],{},[79,15111,15112],{},"γ",[79,15114,2933],{},[137,15116,373],{"stretchy":367},[83,15118,15119],{"encoding":85},"N(x) = \\sum_{i=1}^{n} \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[59,15121,15123,15150,15331],{"className":15122,"ariaHidden":90},[89],[59,15124,15126,15129,15132,15135,15138,15141,15144,15147],{"className":15125},[94],[59,15127],{"className":15128,"style":386},[98],[59,15130,15063],{"className":15131,"style":12843},[103,104],[59,15133,368],{"className":15134},[393],[59,15136,3432],{"className":15137},[103,104],[59,15139,373],{"className":15140},[400],[59,15142],{"className":15143,"style":871},[870],[59,15145,791],{"className":15146},[875],[59,15148],{"className":15149,"style":871},[870],[59,15151,15153,15157,15226,15229,15271,15274,15277,15319,15322,15325,15328],{"className":15152},[94],[59,15154],{"className":15155,"style":15156},[98],"height:2.9291em;vertical-align:-1.2777em;",[59,15158,15160],{"className":15159},[6688,6689],[59,15161,15163,15217],{"className":15162},[166,527],[59,15164,15166,15214],{"className":15165},[170],[59,15167,15169,15190,15200],{"className":15168,"style":6699},[174],[59,15170,15172,15175],{"style":15171},"top:-1.8723em;margin-left:0em;",[59,15173],{"className":15174,"style":6706},[181],[59,15176,15178],{"className":15177},[186,187,188,189],[59,15179,15181,15184,15187],{"className":15180},[103,189],[59,15182,2933],{"className":15183},[103,104,189],[59,15185,791],{"className":15186},[875,189],[59,15188,637],{"className":15189},[103,189],[59,15191,15192,15195],{"style":6724},[59,15193],{"className":15194,"style":6706},[181],[59,15196,15197],{},[59,15198,6527],{"className":15199},[6688,6733,6734],[59,15201,15202,15205],{"style":6737},[59,15203],{"className":15204,"style":6706},[181],[59,15206,15208],{"className":15207},[186,187,188,189],[59,15209,15211],{"className":15210},[103,189],[59,15212,2119],{"className":15213},[103,104,189],[59,15215,564],{"className":15216},[563],[59,15218,15220],{"className":15219},[170],[59,15221,15224],{"className":15222,"style":15223},[174],"height:1.2777em;",[59,15225],{},[59,15227],{"className":15228,"style":983},[870],[59,15230,15232,15236],{"className":15231},[103],[59,15233,15090],{"className":15234,"style":15235},[103,104],"margin-right:0.0037em;",[59,15237,15239],{"className":15238},[162],[59,15240,15242,15263],{"className":15241},[166,527],[59,15243,15245,15260],{"className":15244},[170],[59,15246,15248],{"className":15247,"style":2968},[174],[59,15249,15251,15254],{"style":15250},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[59,15252],{"className":15253,"style":182},[181],[59,15255,15257],{"className":15256},[186,187,188,189],[59,15258,2933],{"className":15259},[103,104,189],[59,15261,564],{"className":15262},[563],[59,15264,15266],{"className":15265},[170],[59,15267,15269],{"className":15268,"style":3092},[174],[59,15270],{},[59,15272,2088],{"className":15273,"style":314},[103,104],[59,15275,368],{"className":15276},[393],[59,15278,15280,15284],{"className":15279},[103],[59,15281,15101],{"className":15282,"style":15283},[103,104],"margin-right:0.0528em;",[59,15285,15287],{"className":15286},[162],[59,15288,15290,15311],{"className":15289},[166,527],[59,15291,15293,15308],{"className":15292},[170],[59,15294,15296],{"className":15295,"style":2968},[174],[59,15297,15299,15302],{"style":15298},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[59,15300],{"className":15301,"style":182},[181],[59,15303,15305],{"className":15304},[186,187,188,189],[59,15306,2933],{"className":15307},[103,104,189],[59,15309,564],{"className":15310},[563],[59,15312,15314],{"className":15313},[170],[59,15315,15317],{"className":15316,"style":3092},[174],[59,15318],{},[59,15320,3432],{"className":15321},[103,104],[59,15323],{"className":15324,"style":105},[870],[59,15326,2000],{"className":15327},[193],[59,15329],{"className":15330,"style":105},[870],[59,15332,15334,15337,15378],{"className":15333},[94],[59,15335],{"className":15336,"style":386},[98],[59,15338,15340,15343],{"className":15339},[103],[59,15341,15112],{"className":15342,"style":4119},[103,104],[59,15344,15346],{"className":15345},[162],[59,15347,15349,15370],{"className":15348},[166,527],[59,15350,15352,15367],{"className":15351},[170],[59,15353,15355],{"className":15354,"style":2968},[174],[59,15356,15358,15361],{"style":15357},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[59,15359],{"className":15360,"style":182},[181],[59,15362,15364],{"className":15363},[186,187,188,189],[59,15365,2933],{"className":15366},[103,104,189],[59,15368,564],{"className":15369},[563],[59,15371,15373],{"className":15372},[170],[59,15374,15376],{"className":15375,"style":3092},[174],[59,15377],{},[59,15379,373],{"className":15380},[400],[14,15382,15383],{},"使得",[59,15385,15387],{"className":15386,"translate":63},[767],[59,15388,15390,15448],{"className":15389,"translate":63},[62],[59,15391,15393],{"className":15392},[67],[69,15394,15395],{"xmlns":71,"display":776},[73,15396,15397,15445],{},[76,15398,15399,15401,15403,15405,15407,15409,15411,15413,15415,15417,15419,15421,15423,15425,15427,15429,15431,15433,15435,15437,15439,15441,15443],{},[79,15400,6298],{"mathvariant":1091},[79,15402,15063],{},[137,15404,368],{"stretchy":367},[79,15406,3432],{},[137,15408,373],{"stretchy":367},[137,15410,6309],{},[79,15412,6010],{},[137,15414,368],{"stretchy":367},[79,15416,3432],{},[137,15418,373],{"stretchy":367},[79,15420,6298],{"mathvariant":1091},[137,15422,6322],{},[79,15424,15007],{},[137,15426,1345],{"separator":90},[870,15428],{"width":461},[79,15430,6332],{"mathvariant":1091},[79,15432,3432],{},[137,15434,6013],{},[137,15436,455],{"stretchy":367},[79,15438,8501],{},[137,15440,1345],{"separator":90},[79,15442,14919],{},[137,15444,487],{"stretchy":367},[83,15446,15447],{"encoding":85},"|N(x) - f(x)| \u003C \\epsilon, \\quad \\forall x\\in [a,b]",[59,15449,15451,15481,15511,15545],{"className":15450,"ariaHidden":90},[89],[59,15452,15454,15457,15460,15463,15466,15469,15472,15475,15478],{"className":15453},[94],[59,15455],{"className":15456,"style":386},[98],[59,15458,6298],{"className":15459},[103],[59,15461,15063],{"className":15462,"style":12843},[103,104],[59,15464,368],{"className":15465},[393],[59,15467,3432],{"className":15468},[103,104],[59,15470,373],{"className":15471},[400],[59,15473],{"className":15474,"style":105},[870],[59,15476,6309],{"className":15477},[193],[59,15479],{"className":15480,"style":105},[870],[59,15482,15484,15487,15490,15493,15496,15499,15502,15505,15508],{"className":15483},[94],[59,15485],{"className":15486,"style":386},[98],[59,15488,6010],{"className":15489,"style":6047},[103,104],[59,15491,368],{"className":15492},[393],[59,15494,3432],{"className":15495},[103,104],[59,15497,373],{"className":15498},[400],[59,15500,6298],{"className":15501},[103],[59,15503],{"className":15504,"style":871},[870],[59,15506,6322],{"className":15507},[875],[59,15509],{"className":15510,"style":871},[870],[59,15512,15514,15517,15520,15523,15527,15530,15533,15536,15539,15542],{"className":15513},[94],[59,15515],{"className":15516,"style":6043},[98],[59,15518,15007],{"className":15519},[103,104],[59,15521,1345],{"className":15522},[1372],[59,15524],{"className":15525,"style":15526},[870],"margin-right:1em;",[59,15528],{"className":15529,"style":983},[870],[59,15531,6332],{"className":15532},[103],[59,15534,3432],{"className":15535},[103,104],[59,15537],{"className":15538,"style":871},[870],[59,15540,6013],{"className":15541},[875],[59,15543],{"className":15544,"style":871},[870],[59,15546,15548,15551,15554,15557,15560,15563,15566],{"className":15547},[94],[59,15549],{"className":15550,"style":386},[98],[59,15552,455],{"className":15553},[393],[59,15555,8501],{"className":15556},[103,104],[59,15558,1345],{"className":15559},[1372],[59,15561],{"className":15562,"style":983},[870],[59,15564,14919],{"className":15565},[103,104],[59,15567,487],{"className":15568},[400],[14,15570,15571,15572,15600],{},"其中 ",[59,15573,15575,15588],{"className":15574,"translate":63},[62],[59,15576,15578],{"className":15577},[67],[69,15579,15580],{"xmlns":71},[73,15581,15582,15586],{},[76,15583,15584],{},[79,15585,2088],{},[83,15587,2091],{"encoding":85},[59,15589,15591],{"className":15590,"ariaHidden":90},[89],[59,15592,15594,15597],{"className":15593},[94],[59,15595],{"className":15596,"style":310},[98],[59,15598,2088],{"className":15599,"style":314},[103,104]," 是一个「非多项式、连续」的激活函数（例如 sigmoid）。",[14,15602,15603],{},"第 1 步：使用魏尔斯特拉斯定理",[14,15605,15606,15607,15650],{},"我们前面已经证明过这个定理：任何连续函数都可以被多项式 ",[59,15608,15610,15629],{"className":15609,"translate":63},[62],[59,15611,15613],{"className":15612},[67],[69,15614,15615],{"xmlns":71},[73,15616,15617,15627],{},[76,15618,15619,15621,15623,15625],{},[79,15620,6249],{},[137,15622,368],{"stretchy":367},[79,15624,3432],{},[137,15626,373],{"stretchy":367},[83,15628,6258],{"encoding":85},[59,15630,15632],{"className":15631,"ariaHidden":90},[89],[59,15633,15635,15638,15641,15644,15647],{"className":15634},[94],[59,15636],{"className":15637,"style":386},[98],[59,15639,6249],{"className":15640,"style":1412},[103,104],[59,15642,368],{"className":15643},[393],[59,15645,3432],{"className":15646},[103,104],[59,15648,373],{"className":15649},[400]," 逼近。即",[59,15652,15654],{"className":15653,"translate":63},[767],[59,15655,15657,15723],{"className":15656,"translate":63},[62],[59,15658,15660],{"className":15659},[67],[69,15661,15662],{"xmlns":71,"display":776},[73,15663,15664,15720],{},[76,15665,15666,15668,15670,15672,15674,15676,15679,15681,15683,15685,15687,15690,15692,15694,15696,15698,15700,15702,15704,15706,15708,15710,15712,15714,15716,15718],{},[79,15667,6332],{"mathvariant":1091},[79,15669,15007],{},[137,15671,6195],{},[473,15673,471],{},[137,15675,1345],{"separator":90},[79,15677,15678],{"mathvariant":1091},"∃",[79,15680,6249],{},[137,15682,368],{"stretchy":367},[79,15684,3432],{},[137,15686,373],{"stretchy":367},[1991,15688,15689],{}," 使得 ",[79,15691,6298],{"mathvariant":1091},[79,15693,6010],{},[137,15695,368],{"stretchy":367},[79,15697,3432],{},[137,15699,373],{"stretchy":367},[137,15701,6309],{},[79,15703,6249],{},[137,15705,368],{"stretchy":367},[79,15707,3432],{},[137,15709,373],{"stretchy":367},[79,15711,6298],{"mathvariant":1091},[137,15713,6322],{},[79,15715,15007],{},[79,15717,1092],{"mathvariant":1091},[473,15719,475],{},[83,15721,15722],{"encoding":85},"\\forall \\epsilon>0, \\exists P(x) \\text{ 使得 } |f(x)-P(x)|\u003C\\epsilon\u002F2",[59,15724,15726,15747,15813,15843],{"className":15725,"ariaHidden":90},[89],[59,15727,15729,15732,15735,15738,15741,15744],{"className":15728},[94],[59,15730],{"className":15731,"style":11071},[98],[59,15733,6332],{"className":15734},[103],[59,15736,15007],{"className":15737},[103,104],[59,15739],{"className":15740,"style":871},[870],[59,15742,6195],{"className":15743},[875],[59,15745],{"className":15746,"style":871},[870],[59,15748,15750,15753,15756,15759,15762,15765,15768,15771,15774,15777,15789,15792,15795,15798,15801,15804,15807,15810],{"className":15749},[94],[59,15751],{"className":15752,"style":386},[98],[59,15754,471],{"className":15755},[103],[59,15757,1345],{"className":15758},[1372],[59,15760],{"className":15761,"style":983},[870],[59,15763,15678],{"className":15764},[103],[59,15766,6249],{"className":15767,"style":1412},[103,104],[59,15769,368],{"className":15770},[393],[59,15772,3432],{"className":15773},[103,104],[59,15775,373],{"className":15776},[400],[59,15778,15780,15783,15786],{"className":15779},[103,2018],[59,15781,3861],{"className":15782},[103],[59,15784,15383],{"className":15785},[103,2022],[59,15787,3861],{"className":15788},[103],[59,15790,6298],{"className":15791},[103],[59,15793,6010],{"className":15794,"style":6047},[103,104],[59,15796,368],{"className":15797},[393],[59,15799,3432],{"className":15800},[103,104],[59,15802,373],{"className":15803},[400],[59,15805],{"className":15806,"style":105},[870],[59,15808,6309],{"className":15809},[193],[59,15811],{"className":15812,"style":105},[870],[59,15814,15816,15819,15822,15825,15828,15831,15834,15837,15840],{"className":15815},[94],[59,15817],{"className":15818,"style":386},[98],[59,15820,6249],{"className":15821,"style":1412},[103,104],[59,15823,368],{"className":15824},[393],[59,15826,3432],{"className":15827},[103,104],[59,15829,373],{"className":15830},[400],[59,15832,6298],{"className":15833},[103],[59,15835],{"className":15836,"style":871},[870],[59,15838,6322],{"className":15839},[875],[59,15841],{"className":15842,"style":871},[870],[59,15844,15846,15849,15852],{"className":15845},[94],[59,15847],{"className":15848,"style":386},[98],[59,15850,15007],{"className":15851},[103,104],[59,15853,15855],{"className":15854},[103],"\u002F2",[14,15857,15858,15859,47],{},"这里我们暂不考虑神经网络，而是把问题转化为「逼近一个多项式」。我们知道，多项式可以写成 ",[59,15860,15862,15908],{"className":15861,"translate":63},[62],[59,15863,15865],{"className":15864},[67],[69,15866,15867],{"xmlns":71},[73,15868,15869,15905],{},[76,15870,15871,15877,15879,15885,15887,15889,15891,15893,15899],{},[2924,15872,15873,15875],{},[79,15874,8501],{},[473,15876,471],{},[137,15878,2000],{},[2924,15880,15881,15883],{},[79,15882,8501],{},[473,15884,637],{},[79,15886,3432],{},[137,15888,2000],{},[137,15890,1513],{},[137,15892,2000],{},[2924,15894,15895,15897],{},[79,15896,8501],{},[79,15898,8496],{},[132,15900,15901,15903],{},[79,15902,3432],{},[79,15904,8496],{},[83,15906,15907],{"encoding":85},"a_0 + a_1 x + \\dots + a_m x^m",[59,15909,15911,15968,16026,16045],{"className":15910,"ariaHidden":90},[89],[59,15912,15914,15918,15959,15962,15965],{"className":15913},[94],[59,15915],{"className":15916,"style":15917},[98],"height:0.7333em;vertical-align:-0.15em;",[59,15919,15921,15924],{"className":15920},[103],[59,15922,8501],{"className":15923},[103,104],[59,15925,15927],{"className":15926},[162],[59,15928,15930,15951],{"className":15929},[166,527],[59,15931,15933,15948],{"className":15932},[170],[59,15934,15937],{"className":15935,"style":15936},[174],"height:0.3011em;",[59,15938,15939,15942],{"style":8241},[59,15940],{"className":15941,"style":182},[181],[59,15943,15945],{"className":15944},[186,187,188,189],[59,15946,471],{"className":15947},[103,189],[59,15949,564],{"className":15950},[563],[59,15952,15954],{"className":15953},[170],[59,15955,15957],{"className":15956,"style":3092},[174],[59,15958],{},[59,15960],{"className":15961,"style":105},[870],[59,15963,2000],{"className":15964},[193],[59,15966],{"className":15967,"style":105},[870],[59,15969,15971,15974,16014,16017,16020,16023],{"className":15970},[94],[59,15972],{"className":15973,"style":15917},[98],[59,15975,15977,15980],{"className":15976},[103],[59,15978,8501],{"className":15979},[103,104],[59,15981,15983],{"className":15982},[162],[59,15984,15986,16006],{"className":15985},[166,527],[59,15987,15989,16003],{"className":15988},[170],[59,15990,15992],{"className":15991,"style":15936},[174],[59,15993,15994,15997],{"style":8241},[59,15995],{"className":15996,"style":182},[181],[59,15998,16000],{"className":15999},[186,187,188,189],[59,16001,637],{"className":16002},[103,189],[59,16004,564],{"className":16005},[563],[59,16007,16009],{"className":16008},[170],[59,16010,16012],{"className":16011,"style":3092},[174],[59,16013],{},[59,16015,3432],{"className":16016},[103,104],[59,16018],{"className":16019,"style":105},[870],[59,16021,2000],{"className":16022},[193],[59,16024],{"className":16025,"style":105},[870],[59,16027,16029,16033,16036,16039,16042],{"className":16028},[94],[59,16030],{"className":16031,"style":16032},[98],"height:0.6667em;vertical-align:-0.0833em;",[59,16034,1513],{"className":16035},[504],[59,16037],{"className":16038,"style":105},[870],[59,16040,2000],{"className":16041},[193],[59,16043],{"className":16044,"style":105},[870],[59,16046,16048,16052,16092],{"className":16047},[94],[59,16049],{"className":16050,"style":16051},[98],"height:0.8144em;vertical-align:-0.15em;",[59,16053,16055,16058],{"className":16054},[103],[59,16056,8501],{"className":16057},[103,104],[59,16059,16061],{"className":16060},[162],[59,16062,16064,16084],{"className":16063},[166,527],[59,16065,16067,16081],{"className":16066},[170],[59,16068,16070],{"className":16069,"style":6627},[174],[59,16071,16072,16075],{"style":8241},[59,16073],{"className":16074,"style":182},[181],[59,16076,16078],{"className":16077},[186,187,188,189],[59,16079,8496],{"className":16080},[103,104,189],[59,16082,564],{"className":16083},[563],[59,16085,16087],{"className":16086},[170],[59,16088,16090],{"className":16089,"style":3092},[174],[59,16091],{},[59,16093,16095,16098],{"className":16094},[103],[59,16096,3432],{"className":16097},[103,104],[59,16099,16101],{"className":16100},[162],[59,16102,16104],{"className":16103},[166],[59,16105,16107],{"className":16106},[170],[59,16108,16111],{"className":16109,"style":16110},[174],"height:0.6644em;",[59,16112,16113,16116],{"style":177},[59,16114],{"className":16115,"style":182},[181],[59,16117,16119],{"className":16118},[186,187,188,189],[59,16120,8496],{"className":16121},[103,104,189],[14,16123,16124,16125],{},"第 2 步：用激活函数逼近单项式 ",[59,16126,16128,16146],{"className":16127,"translate":63},[62],[59,16129,16131],{"className":16130},[67],[69,16132,16133],{"xmlns":71},[73,16134,16135,16143],{},[76,16136,16137],{},[132,16138,16139,16141],{},[79,16140,3432],{},[79,16142,3659],{},[83,16144,16145],{"encoding":85},"x^k",[59,16147,16149],{"className":16148,"ariaHidden":90},[89],[59,16150,16152,16155],{"className":16151},[94],[59,16153],{"className":16154,"style":4680},[98],[59,16156,16158,16161],{"className":16157},[103],[59,16159,3432],{"className":16160},[103,104],[59,16162,16164],{"className":16163},[162],[59,16165,16167],{"className":16166},[166],[59,16168,16170],{"className":16169},[170],[59,16171,16173],{"className":16172,"style":4680},[174],[59,16174,16175,16178],{"style":177},[59,16176],{"className":16177,"style":182},[181],[59,16179,16181],{"className":16180},[186,187,188,189],[59,16182,3659],{"className":16183,"style":4157},[103,104,189],[14,16185,16186,16187,16245,16246,47],{},"如果我们能用单隐层中若干神经元的线性组合逼近 ",[59,16188,16190,16207],{"className":16189,"translate":63},[62],[59,16191,16193],{"className":16192},[67],[69,16194,16195],{"xmlns":71},[73,16196,16197,16205],{},[76,16198,16199],{},[132,16200,16201,16203],{},[79,16202,3432],{},[79,16204,3659],{},[83,16206,16145],{"encoding":85},[59,16208,16210],{"className":16209,"ariaHidden":90},[89],[59,16211,16213,16216],{"className":16212},[94],[59,16214],{"className":16215,"style":4680},[98],[59,16217,16219,16222],{"className":16218},[103],[59,16220,3432],{"className":16221},[103,104],[59,16223,16225],{"className":16224},[162],[59,16226,16228],{"className":16227},[166],[59,16229,16231],{"className":16230},[170],[59,16232,16234],{"className":16233,"style":4680},[174],[59,16235,16236,16239],{"style":177},[59,16237],{"className":16238,"style":182},[181],[59,16240,16242],{"className":16241},[186,187,188,189],[59,16243,3659],{"className":16244,"style":4157},[103,104,189],"，那么就能逼近多项式 ",[59,16247,16249,16268],{"className":16248,"translate":63},[62],[59,16250,16252],{"className":16251},[67],[69,16253,16254],{"xmlns":71},[73,16255,16256,16266],{},[76,16257,16258,16260,16262,16264],{},[79,16259,6249],{},[137,16261,368],{"stretchy":367},[79,16263,3432],{},[137,16265,373],{"stretchy":367},[83,16267,6258],{"encoding":85},[59,16269,16271],{"className":16270,"ariaHidden":90},[89],[59,16272,16274,16277,16280,16283,16286],{"className":16273},[94],[59,16275],{"className":16276,"style":386},[98],[59,16278,6249],{"className":16279,"style":1412},[103,104],[59,16281,368],{"className":16282},[393],[59,16284,3432],{"className":16285},[103,104],[59,16287,373],{"className":16288},[400],[14,16290,16291],{},"单隐层网络的形式为：",[59,16293,16295],{"className":16294,"translate":63},[767],[59,16296,16298,16362],{"className":16297,"translate":63},[62],[59,16299,16301],{"className":16300},[67],[69,16302,16303],{"xmlns":71,"display":776},[73,16304,16305,16359],{},[76,16306,16307,16309,16311,16313,16315,16317,16331,16337,16339,16341,16347,16349,16351,16357],{},[79,16308,15063],{},[137,16310,368],{"stretchy":367},[79,16312,3432],{},[137,16314,373],{"stretchy":367},[137,16316,791],{},[6523,16318,16319,16321,16329],{},[137,16320,6527],{},[76,16322,16323,16325,16327],{},[79,16324,2933],{},[137,16326,791],{},[473,16328,637],{},[79,16330,2119],{},[2924,16332,16333,16335],{},[79,16334,15090],{},[79,16336,2933],{},[79,16338,2088],{},[137,16340,368],{"stretchy":367},[2924,16342,16343,16345],{},[79,16344,15101],{},[79,16346,2933],{},[79,16348,3432],{},[137,16350,2000],{},[2924,16352,16353,16355],{},[79,16354,15112],{},[79,16356,2933],{},[137,16358,373],{"stretchy":367},[83,16360,16361],{"encoding":85},"  N(x) = \\sum_{i=1}^{n} \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[59,16363,16365,16392,16566],{"className":16364,"ariaHidden":90},[89],[59,16366,16368,16371,16374,16377,16380,16383,16386,16389],{"className":16367},[94],[59,16369],{"className":16370,"style":386},[98],[59,16372,15063],{"className":16373,"style":12843},[103,104],[59,16375,368],{"className":16376},[393],[59,16378,3432],{"className":16379},[103,104],[59,16381,373],{"className":16382},[400],[59,16384],{"className":16385,"style":871},[870],[59,16387,791],{"className":16388},[875],[59,16390],{"className":16391,"style":871},[870],[59,16393,16395,16398,16465,16468,16508,16511,16514,16554,16557,16560,16563],{"className":16394},[94],[59,16396],{"className":16397,"style":15156},[98],[59,16399,16401],{"className":16400},[6688,6689],[59,16402,16404,16457],{"className":16403},[166,527],[59,16405,16407,16454],{"className":16406},[170],[59,16408,16410,16430,16440],{"className":16409,"style":6699},[174],[59,16411,16412,16415],{"style":15171},[59,16413],{"className":16414,"style":6706},[181],[59,16416,16418],{"className":16417},[186,187,188,189],[59,16419,16421,16424,16427],{"className":16420},[103,189],[59,16422,2933],{"className":16423},[103,104,189],[59,16425,791],{"className":16426},[875,189],[59,16428,637],{"className":16429},[103,189],[59,16431,16432,16435],{"style":6724},[59,16433],{"className":16434,"style":6706},[181],[59,16436,16437],{},[59,16438,6527],{"className":16439},[6688,6733,6734],[59,16441,16442,16445],{"style":6737},[59,16443],{"className":16444,"style":6706},[181],[59,16446,16448],{"className":16447},[186,187,188,189],[59,16449,16451],{"className":16450},[103,189],[59,16452,2119],{"className":16453},[103,104,189],[59,16455,564],{"className":16456},[563],[59,16458,16460],{"className":16459},[170],[59,16461,16463],{"className":16462,"style":15223},[174],[59,16464],{},[59,16466],{"className":16467,"style":983},[870],[59,16469,16471,16474],{"className":16470},[103],[59,16472,15090],{"className":16473,"style":15235},[103,104],[59,16475,16477],{"className":16476},[162],[59,16478,16480,16500],{"className":16479},[166,527],[59,16481,16483,16497],{"className":16482},[170],[59,16484,16486],{"className":16485,"style":2968},[174],[59,16487,16488,16491],{"style":15250},[59,16489],{"className":16490,"style":182},[181],[59,16492,16494],{"className":16493},[186,187,188,189],[59,16495,2933],{"className":16496},[103,104,189],[59,16498,564],{"className":16499},[563],[59,16501,16503],{"className":16502},[170],[59,16504,16506],{"className":16505,"style":3092},[174],[59,16507],{},[59,16509,2088],{"className":16510,"style":314},[103,104],[59,16512,368],{"className":16513},[393],[59,16515,16517,16520],{"className":16516},[103],[59,16518,15101],{"className":16519,"style":15283},[103,104],[59,16521,16523],{"className":16522},[162],[59,16524,16526,16546],{"className":16525},[166,527],[59,16527,16529,16543],{"className":16528},[170],[59,16530,16532],{"className":16531,"style":2968},[174],[59,16533,16534,16537],{"style":15298},[59,16535],{"className":16536,"style":182},[181],[59,16538,16540],{"className":16539},[186,187,188,189],[59,16541,2933],{"className":16542},[103,104,189],[59,16544,564],{"className":16545},[563],[59,16547,16549],{"className":16548},[170],[59,16550,16552],{"className":16551,"style":3092},[174],[59,16553],{},[59,16555,3432],{"className":16556},[103,104],[59,16558],{"className":16559,"style":105},[870],[59,16561,2000],{"className":16562},[193],[59,16564],{"className":16565,"style":105},[870],[59,16567,16569,16572,16612],{"className":16568},[94],[59,16570],{"className":16571,"style":386},[98],[59,16573,16575,16578],{"className":16574},[103],[59,16576,15112],{"className":16577,"style":4119},[103,104],[59,16579,16581],{"className":16580},[162],[59,16582,16584,16604],{"className":16583},[166,527],[59,16585,16587,16601],{"className":16586},[170],[59,16588,16590],{"className":16589,"style":2968},[174],[59,16591,16592,16595],{"style":15357},[59,16593],{"className":16594,"style":182},[181],[59,16596,16598],{"className":16597},[186,187,188,189],[59,16599,2933],{"className":16600},[103,104,189],[59,16602,564],{"className":16603},[563],[59,16605,16607],{"className":16606},[170],[59,16608,16610],{"className":16609,"style":3092},[174],[59,16611],{},[59,16613,373],{"className":16614},[400],[14,16616,16617,16618,16662],{},"选取合适的 ",[59,16619,16621,16641],{"className":16620,"translate":63},[62],[59,16622,16624],{"className":16623},[67],[69,16625,16626],{"xmlns":71},[73,16627,16628,16638],{},[76,16629,16630,16632,16634,16636],{},[79,16631,2088],{},[137,16633,368],{"stretchy":367},[79,16635,3432],{},[137,16637,373],{"stretchy":367},[83,16639,16640],{"encoding":85},"\\sigma(x)",[59,16642,16644],{"className":16643,"ariaHidden":90},[89],[59,16645,16647,16650,16653,16656,16659],{"className":16646},[94],[59,16648],{"className":16649,"style":386},[98],[59,16651,2088],{"className":16652,"style":314},[103,104],[59,16654,368],{"className":16655},[393],[59,16657,3432],{"className":16658},[103,104],[59,16660,373],{"className":16661},[400],"，例如 sigmoid：",[59,16664,16666],{"className":16665,"translate":63},[767],[59,16667,16669,16712],{"className":16668,"translate":63},[62],[59,16670,16672],{"className":16671},[67],[69,16673,16674],{"xmlns":71,"display":776},[73,16675,16676,16709],{},[76,16677,16678,16680,16682,16684,16686,16688],{},[79,16679,2088],{},[137,16681,368],{"stretchy":367},[79,16683,3432],{},[137,16685,373],{"stretchy":367},[137,16687,791],{},[1174,16689,16690,16692],{},[473,16691,637],{},[76,16693,16694,16696,16698],{},[473,16695,637],{},[137,16697,2000],{},[132,16699,16700,16703],{},[79,16701,16702],{},"e",[76,16704,16705,16707],{},[137,16706,6309],{},[79,16708,3432],{},[83,16710,16711],{"encoding":85},"   \\sigma(x) = \\frac{1}{1+e^{-x}}",[59,16713,16715,16742],{"className":16714,"ariaHidden":90},[89],[59,16716,16718,16721,16724,16727,16730,16733,16736,16739],{"className":16717},[94],[59,16719],{"className":16720,"style":386},[98],[59,16722,2088],{"className":16723,"style":314},[103,104],[59,16725,368],{"className":16726},[393],[59,16728,3432],{"className":16729},[103,104],[59,16731,373],{"className":16732},[400],[59,16734],{"className":16735,"style":871},[870],[59,16737,791],{"className":16738},[875],[59,16740],{"className":16741,"style":871},[870],[59,16743,16745,16749],{"className":16744},[94],[59,16746],{"className":16747,"style":16748},[98],"height:2.0908em;vertical-align:-0.7693em;",[59,16750,16752,16755,16854],{"className":16751},[103],[59,16753],{"className":16754},[393,1222],[59,16756,16758],{"className":16757},[1174],[59,16759,16761,16845],{"className":16760},[166,527],[59,16762,16764,16842],{"className":16763},[170],[59,16765,16767,16823,16831],{"className":16766,"style":8045},[174],[59,16768,16769,16772],{"style":4109},[59,16770],{"className":16771,"style":541},[181],[59,16773,16775,16778,16781,16784,16787],{"className":16774},[103],[59,16776,637],{"className":16777},[103],[59,16779],{"className":16780,"style":105},[870],[59,16782,2000],{"className":16783},[193],[59,16785],{"className":16786,"style":105},[870],[59,16788,16790,16793],{"className":16789},[103],[59,16791,16702],{"className":16792},[103,104],[59,16794,16796],{"className":16795},[162],[59,16797,16799],{"className":16798},[166],[59,16800,16802],{"className":16801},[170],[59,16803,16806],{"className":16804,"style":16805},[174],"height:0.6973em;",[59,16807,16808,16811],{"style":4141},[59,16809],{"className":16810,"style":182},[181],[59,16812,16814],{"className":16813},[186,187,188,189],[59,16815,16817,16820],{"className":16816},[103,189],[59,16818,6309],{"className":16819},[103,189],[59,16821,3432],{"className":16822},[103,104,189],[59,16824,16825,16828],{"style":1253},[59,16826],{"className":16827,"style":541},[181],[59,16829],{"className":16830,"style":1261},[1260],[59,16832,16833,16836],{"style":4203},[59,16834],{"className":16835,"style":541},[181],[59,16837,16839],{"className":16838},[103],[59,16840,637],{"className":16841},[103],[59,16843,564],{"className":16844},[563],[59,16846,16848],{"className":16847},[170],[59,16849,16852],{"className":16850,"style":16851},[174],"height:0.7693em;",[59,16853],{},[59,16855],{"className":16856},[400,1222],[14,16858,16859],{},"它的泰勒展开在局部非零：",[59,16861,16863],{"className":16862,"translate":63},[767],[59,16864,16866,16923],{"className":16865,"translate":63},[62],[59,16867,16869],{"className":16868},[67],[69,16870,16871],{"xmlns":71,"display":776},[73,16872,16873,16920],{},[76,16874,16875,16877,16879,16881,16883,16885,16891,16893,16899,16901,16903,16910,16916,16918],{},[79,16876,2088],{},[137,16878,368],{"stretchy":367},[79,16880,3432],{},[137,16882,373],{"stretchy":367},[137,16884,791],{},[1174,16886,16887,16889],{},[473,16888,637],{},[473,16890,475],{},[137,16892,2000],{},[1174,16894,16895,16897],{},[473,16896,637],{},[473,16898,644],{},[79,16900,3432],{},[137,16902,6309],{},[1174,16904,16905,16907],{},[473,16906,637],{},[473,16908,16909],{},"48",[132,16911,16912,16914],{},[79,16913,3432],{},[473,16915,484],{},[137,16917,2000],{},[137,16919,7971],{},[83,16921,16922],{"encoding":85},"  \\sigma(x) = \\frac12 + \\frac14 x - \\frac1{48} x^3 + \\dots",[59,16924,16926,16953,17030,17110,17217],{"className":16925,"ariaHidden":90},[89],[59,16927,16929,16932,16935,16938,16941,16944,16947,16950],{"className":16928},[94],[59,16930],{"className":16931,"style":386},[98],[59,16933,2088],{"className":16934,"style":314},[103,104],[59,16936,368],{"className":16937},[393],[59,16939,3432],{"className":16940},[103,104],[59,16942,373],{"className":16943},[400],[59,16945],{"className":16946,"style":871},[870],[59,16948,791],{"className":16949},[875],[59,16951],{"className":16952,"style":871},[870],[59,16954,16956,16959,17021,17024,17027],{"className":16955},[94],[59,16957],{"className":16958,"style":13624},[98],[59,16960,16962,16965,17018],{"className":16961},[103],[59,16963],{"className":16964},[393,1222],[59,16966,16968],{"className":16967},[1174],[59,16969,16971,17010],{"className":16970},[166,527],[59,16972,16974,17007],{"className":16973},[170],[59,16975,16977,16988,16996],{"className":16976,"style":8045},[174],[59,16978,16979,16982],{"style":4109},[59,16980],{"className":16981,"style":541},[181],[59,16983,16985],{"className":16984},[103],[59,16986,475],{"className":16987},[103],[59,16989,16990,16993],{"style":1253},[59,16991],{"className":16992,"style":541},[181],[59,16994],{"className":16995,"style":1261},[1260],[59,16997,16998,17001],{"style":4203},[59,16999],{"className":17000,"style":541},[181],[59,17002,17004],{"className":17003},[103],[59,17005,637],{"className":17006},[103],[59,17008,564],{"className":17009},[563],[59,17011,17013],{"className":17012},[170],[59,17014,17016],{"className":17015,"style":4298},[174],[59,17017],{},[59,17019],{"className":17020},[400,1222],[59,17022],{"className":17023,"style":105},[870],[59,17025,2000],{"className":17026},[193],[59,17028],{"className":17029,"style":105},[870],[59,17031,17033,17036,17098,17101,17104,17107],{"className":17032},[94],[59,17034],{"className":17035,"style":13624},[98],[59,17037,17039,17042,17095],{"className":17038},[103],[59,17040],{"className":17041},[393,1222],[59,17043,17045],{"className":17044},[1174],[59,17046,17048,17087],{"className":17047},[166,527],[59,17049,17051,17084],{"className":17050},[170],[59,17052,17054,17065,17073],{"className":17053,"style":8045},[174],[59,17055,17056,17059],{"style":4109},[59,17057],{"className":17058,"style":541},[181],[59,17060,17062],{"className":17061},[103],[59,17063,644],{"className":17064},[103],[59,17066,17067,17070],{"style":1253},[59,17068],{"className":17069,"style":541},[181],[59,17071],{"className":17072,"style":1261},[1260],[59,17074,17075,17078],{"style":4203},[59,17076],{"className":17077,"style":541},[181],[59,17079,17081],{"className":17080},[103],[59,17082,637],{"className":17083},[103],[59,17085,564],{"className":17086},[563],[59,17088,17090],{"className":17089},[170],[59,17091,17093],{"className":17092,"style":4298},[174],[59,17094],{},[59,17096],{"className":17097},[400,1222],[59,17099,3432],{"className":17100},[103,104],[59,17102],{"className":17103,"style":105},[870],[59,17105,6309],{"className":17106},[193],[59,17108],{"className":17109,"style":105},[870],[59,17111,17113,17116,17178,17208,17211,17214],{"className":17112},[94],[59,17114],{"className":17115,"style":13624},[98],[59,17117,17119,17122,17175],{"className":17118},[103],[59,17120],{"className":17121},[393,1222],[59,17123,17125],{"className":17124},[1174],[59,17126,17128,17167],{"className":17127},[166,527],[59,17129,17131,17164],{"className":17130},[170],[59,17132,17134,17145,17153],{"className":17133,"style":8045},[174],[59,17135,17136,17139],{"style":4109},[59,17137],{"className":17138,"style":541},[181],[59,17140,17142],{"className":17141},[103],[59,17143,16909],{"className":17144},[103],[59,17146,17147,17150],{"style":1253},[59,17148],{"className":17149,"style":541},[181],[59,17151],{"className":17152,"style":1261},[1260],[59,17154,17155,17158],{"style":4203},[59,17156],{"className":17157,"style":541},[181],[59,17159,17161],{"className":17160},[103],[59,17162,637],{"className":17163},[103],[59,17165,564],{"className":17166},[563],[59,17168,17170],{"className":17169},[170],[59,17171,17173],{"className":17172,"style":4298},[174],[59,17174],{},[59,17176],{"className":17177},[400,1222],[59,17179,17181,17184],{"className":17180},[103],[59,17182,3432],{"className":17183},[103,104],[59,17185,17187],{"className":17186},[162],[59,17188,17190],{"className":17189},[166],[59,17191,17193],{"className":17192},[170],[59,17194,17197],{"className":17195,"style":17196},[174],"height:0.8641em;",[59,17198,17199,17202],{"style":1711},[59,17200],{"className":17201,"style":182},[181],[59,17203,17205],{"className":17204},[186,187,188,189],[59,17206,484],{"className":17207},[103,189],[59,17209],{"className":17210,"style":105},[870],[59,17212,2000],{"className":17213},[193],[59,17215],{"className":17216,"style":105},[870],[59,17218,17220,17224],{"className":17219},[94],[59,17221],{"className":17222,"style":17223},[98],"height:0.123em;",[59,17225,7971],{"className":17226},[504],[14,17228,17229],{},"利用其平移版本的线性组合，可以生成任意的多项式项。",[14,17231,17232,17233,17411],{},"通过调节 ",[59,17234,17236,17270],{"className":17235,"translate":63},[62],[59,17237,17239],{"className":17238},[67],[69,17240,17241],{"xmlns":71},[73,17242,17243,17267],{},[76,17244,17245,17251,17253,17259,17261],{},[2924,17246,17247,17249],{},[79,17248,15090],{},[79,17250,2933],{},[137,17252,1345],{"separator":90},[2924,17254,17255,17257],{},[79,17256,15101],{},[79,17258,2933],{},[137,17260,1345],{"separator":90},[2924,17262,17263,17265],{},[79,17264,15112],{},[79,17266,2933],{},[83,17268,17269],{"encoding":85},"\\alpha_i, \\beta_i, \\gamma_i",[59,17271,17273],{"className":17272,"ariaHidden":90},[89],[59,17274,17276,17279,17319,17322,17325,17365,17368,17371],{"className":17275},[94],[59,17277],{"className":17278,"style":6043},[98],[59,17280,17282,17285],{"className":17281},[103],[59,17283,15090],{"className":17284,"style":15235},[103,104],[59,17286,17288],{"className":17287},[162],[59,17289,17291,17311],{"className":17290},[166,527],[59,17292,17294,17308],{"className":17293},[170],[59,17295,17297],{"className":17296,"style":2968},[174],[59,17298,17299,17302],{"style":15250},[59,17300],{"className":17301,"style":182},[181],[59,17303,17305],{"className":17304},[186,187,188,189],[59,17306,2933],{"className":17307},[103,104,189],[59,17309,564],{"className":17310},[563],[59,17312,17314],{"className":17313},[170],[59,17315,17317],{"className":17316,"style":3092},[174],[59,17318],{},[59,17320,1345],{"className":17321},[1372],[59,17323],{"className":17324,"style":983},[870],[59,17326,17328,17331],{"className":17327},[103],[59,17329,15101],{"className":17330,"style":15283},[103,104],[59,17332,17334],{"className":17333},[162],[59,17335,17337,17357],{"className":17336},[166,527],[59,17338,17340,17354],{"className":17339},[170],[59,17341,17343],{"className":17342,"style":2968},[174],[59,17344,17345,17348],{"style":15298},[59,17346],{"className":17347,"style":182},[181],[59,17349,17351],{"className":17350},[186,187,188,189],[59,17352,2933],{"className":17353},[103,104,189],[59,17355,564],{"className":17356},[563],[59,17358,17360],{"className":17359},[170],[59,17361,17363],{"className":17362,"style":3092},[174],[59,17364],{},[59,17366,1345],{"className":17367},[1372],[59,17369],{"className":17370,"style":983},[870],[59,17372,17374,17377],{"className":17373},[103],[59,17375,15112],{"className":17376,"style":4119},[103,104],[59,17378,17380],{"className":17379},[162],[59,17381,17383,17403],{"className":17382},[166,527],[59,17384,17386,17400],{"className":17385},[170],[59,17387,17389],{"className":17388,"style":2968},[174],[59,17390,17391,17394],{"style":15357},[59,17392],{"className":17393,"style":182},[181],[59,17395,17397],{"className":17396},[186,187,188,189],[59,17398,2933],{"className":17399},[103,104,189],[59,17401,564],{"className":17402},[563],[59,17404,17406],{"className":17405},[170],[59,17407,17409],{"className":17408,"style":3092},[174],[59,17410],{},"，线性组合",[59,17413,17415],{"className":17414,"translate":63},[767],[59,17416,17418,17464],{"className":17417,"translate":63},[62],[59,17419,17421],{"className":17420},[67],[69,17422,17423],{"xmlns":71,"display":776},[73,17424,17425,17461],{},[76,17426,17427,17433,17439,17441,17443,17449,17451,17453,17459],{},[1497,17428,17429,17431],{},[137,17430,6527],{},[79,17432,2933],{},[2924,17434,17435,17437],{},[79,17436,15090],{},[79,17438,2933],{},[79,17440,2088],{},[137,17442,368],{"stretchy":367},[2924,17444,17445,17447],{},[79,17446,15101],{},[79,17448,2933],{},[79,17450,3432],{},[137,17452,2000],{},[2924,17454,17455,17457],{},[79,17456,15112],{},[79,17458,2933],{},[137,17460,373],{"stretchy":367},[83,17462,17463],{"encoding":85},"   \\sum_i \\alpha_i \\sigma(\\beta_i x + \\gamma_i)",[59,17465,17467,17620],{"className":17466,"ariaHidden":90},[89],[59,17468,17470,17474,17519,17522,17562,17565,17568,17608,17611,17614,17617],{"className":17469},[94],[59,17471],{"className":17472,"style":17473},[98],"height:2.3277em;vertical-align:-1.2777em;",[59,17475,17477],{"className":17476},[6688,6689],[59,17478,17480,17511],{"className":17479},[166,527],[59,17481,17483,17508],{"className":17482},[170],[59,17484,17487,17498],{"className":17485,"style":17486},[174],"height:1.05em;",[59,17488,17489,17492],{"style":15171},[59,17490],{"className":17491,"style":6706},[181],[59,17493,17495],{"className":17494},[186,187,188,189],[59,17496,2933],{"className":17497},[103,104,189],[59,17499,17500,17503],{"style":6724},[59,17501],{"className":17502,"style":6706},[181],[59,17504,17505],{},[59,17506,6527],{"className":17507},[6688,6733,6734],[59,17509,564],{"className":17510},[563],[59,17512,17514],{"className":17513},[170],[59,17515,17517],{"className":17516,"style":15223},[174],[59,17518],{},[59,17520],{"className":17521,"style":983},[870],[59,17523,17525,17528],{"className":17524},[103],[59,17526,15090],{"className":17527,"style":15235},[103,104],[59,17529,17531],{"className":17530},[162],[59,17532,17534,17554],{"className":17533},[166,527],[59,17535,17537,17551],{"className":17536},[170],[59,17538,17540],{"className":17539,"style":2968},[174],[59,17541,17542,17545],{"style":15250},[59,17543],{"className":17544,"style":182},[181],[59,17546,17548],{"className":17547},[186,187,188,189],[59,17549,2933],{"className":17550},[103,104,189],[59,17552,564],{"className":17553},[563],[59,17555,17557],{"className":17556},[170],[59,17558,17560],{"className":17559,"style":3092},[174],[59,17561],{},[59,17563,2088],{"className":17564,"style":314},[103,104],[59,17566,368],{"className":17567},[393],[59,17569,17571,17574],{"className":17570},[103],[59,17572,15101],{"className":17573,"style":15283},[103,104],[59,17575,17577],{"className":17576},[162],[59,17578,17580,17600],{"className":17579},[166,527],[59,17581,17583,17597],{"className":17582},[170],[59,17584,17586],{"className":17585,"style":2968},[174],[59,17587,17588,17591],{"style":15298},[59,17589],{"className":17590,"style":182},[181],[59,17592,17594],{"className":17593},[186,187,188,189],[59,17595,2933],{"className":17596},[103,104,189],[59,17598,564],{"className":17599},[563],[59,17601,17603],{"className":17602},[170],[59,17604,17606],{"className":17605,"style":3092},[174],[59,17607],{},[59,17609,3432],{"className":17610},[103,104],[59,17612],{"className":17613,"style":105},[870],[59,17615,2000],{"className":17616},[193],[59,17618],{"className":17619,"style":105},[870],[59,17621,17623,17626,17666],{"className":17622},[94],[59,17624],{"className":17625,"style":386},[98],[59,17627,17629,17632],{"className":17628},[103],[59,17630,15112],{"className":17631,"style":4119},[103,104],[59,17633,17635],{"className":17634},[162],[59,17636,17638,17658],{"className":17637},[166,527],[59,17639,17641,17655],{"className":17640},[170],[59,17642,17644],{"className":17643,"style":2968},[174],[59,17645,17646,17649],{"style":15357},[59,17647],{"className":17648,"style":182},[181],[59,17650,17652],{"className":17651},[186,187,188,189],[59,17653,2933],{"className":17654},[103,104,189],[59,17656,564],{"className":17657},[563],[59,17659,17661],{"className":17660},[170],[59,17662,17664],{"className":17663,"style":3092},[174],[59,17665],{},[59,17667,373],{"className":17668},[400],[14,17670,17671,17672,17730],{},"可以任意精度逼近 ",[59,17673,17675,17692],{"className":17674,"translate":63},[62],[59,17676,17678],{"className":17677},[67],[69,17679,17680],{"xmlns":71},[73,17681,17682,17690],{},[76,17683,17684],{},[132,17685,17686,17688],{},[79,17687,3432],{},[79,17689,3659],{},[83,17691,16145],{"encoding":85},[59,17693,17695],{"className":17694,"ariaHidden":90},[89],[59,17696,17698,17701],{"className":17697},[94],[59,17699],{"className":17700,"style":4680},[98],[59,17702,17704,17707],{"className":17703},[103],[59,17705,3432],{"className":17706},[103,104],[59,17708,17710],{"className":17709},[162],[59,17711,17713],{"className":17712},[166],[59,17714,17716],{"className":17715},[170],[59,17717,17719],{"className":17718,"style":4680},[174],[59,17720,17721,17724],{"style":177},[59,17722],{"className":17723,"style":182},[181],[59,17725,17727],{"className":17726},[186,187,188,189],[59,17728,3659],{"className":17729,"style":4157},[103,104,189],"（类似于微积分中的泰勒多项式逼近）。",[11,17732,17733],{},[14,17734,17735],{},"这里的要点直觉：每个神经元都是一个「非线性函数块」，把许多这样的块加在一起，就能组合成多项式形式。",[14,17737,17738],{},"第 3 步：组合逼近多项式",[14,17740,17741,17742,17968,17969,18027,18028,18114],{},"对多项式 ",[59,17743,17745,17793],{"className":17744,"translate":63},[62],[59,17746,17748],{"className":17747},[67],[69,17749,17750],{"xmlns":71},[73,17751,17752,17790],{},[76,17753,17754,17756,17758,17760,17762,17764,17778,17784],{},[79,17755,6249],{},[137,17757,368],{"stretchy":367},[79,17759,3432],{},[137,17761,373],{"stretchy":367},[137,17763,791],{},[3581,17765,17766,17768,17776],{},[137,17767,6527],{},[76,17769,17770,17772,17774],{},[79,17771,3659],{},[137,17773,791],{},[473,17775,471],{},[79,17777,8496],{},[2924,17779,17780,17782],{},[79,17781,8501],{},[79,17783,3659],{},[132,17785,17786,17788],{},[79,17787,3432],{},[79,17789,3659],{},[83,17791,17792],{"encoding":85},"P(x) = \\sum_{k=0}^m a_k x^k",[59,17794,17796,17823],{"className":17795,"ariaHidden":90},[89],[59,17797,17799,17802,17805,17808,17811,17814,17817,17820],{"className":17798},[94],[59,17800],{"className":17801,"style":386},[98],[59,17803,6249],{"className":17804,"style":1412},[103,104],[59,17806,368],{"className":17807},[393],[59,17809,3432],{"className":17810},[103,104],[59,17812,373],{"className":17813},[400],[59,17815],{"className":17816,"style":871},[870],[59,17818,791],{"className":17819},[875],[59,17821],{"className":17822,"style":871},[870],[59,17824,17826,17830,17896,17899,17939],{"className":17825},[94],[59,17827],{"className":17828,"style":17829},[98],"height:1.1488em;vertical-align:-0.2997em;",[59,17831,17833,17838],{"className":17832},[6688],[59,17834,6527],{"className":17835,"style":17837},[6688,6733,17836],"small-op","position:relative;top:0em;",[59,17839,17841],{"className":17840},[162],[59,17842,17844,17887],{"className":17843},[166,527],[59,17845,17847,17884],{"className":17846},[170],[59,17848,17851,17872],{"className":17849,"style":17850},[174],"height:0.8043em;",[59,17852,17854,17857],{"style":17853},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[59,17855],{"className":17856,"style":182},[181],[59,17858,17860],{"className":17859},[186,187,188,189],[59,17861,17863,17866,17869],{"className":17862},[103,189],[59,17864,3659],{"className":17865,"style":4157},[103,104,189],[59,17867,791],{"className":17868},[875,189],[59,17870,471],{"className":17871},[103,189],[59,17873,17875,17878],{"style":17874},"top:-3.2029em;margin-right:0.05em;",[59,17876],{"className":17877,"style":182},[181],[59,17879,17881],{"className":17880},[186,187,188,189],[59,17882,8496],{"className":17883},[103,104,189],[59,17885,564],{"className":17886},[563],[59,17888,17890],{"className":17889},[170],[59,17891,17894],{"className":17892,"style":17893},[174],"height:0.2997em;",[59,17895],{},[59,17897],{"className":17898,"style":983},[870],[59,17900,17902,17905],{"className":17901},[103],[59,17903,8501],{"className":17904},[103,104],[59,17906,17908],{"className":17907},[162],[59,17909,17911,17931],{"className":17910},[166,527],[59,17912,17914,17928],{"className":17913},[170],[59,17915,17917],{"className":17916,"style":8238},[174],[59,17918,17919,17922],{"style":8241},[59,17920],{"className":17921,"style":182},[181],[59,17923,17925],{"className":17924},[186,187,188,189],[59,17926,3659],{"className":17927,"style":4157},[103,104,189],[59,17929,564],{"className":17930},[563],[59,17932,17934],{"className":17933},[170],[59,17935,17937],{"className":17936,"style":3092},[174],[59,17938],{},[59,17940,17942,17945],{"className":17941},[103],[59,17943,3432],{"className":17944},[103,104],[59,17946,17948],{"className":17947},[162],[59,17949,17951],{"className":17950},[166],[59,17952,17954],{"className":17953},[170],[59,17955,17957],{"className":17956,"style":4680},[174],[59,17958,17959,17962],{"style":177},[59,17960],{"className":17961,"style":182},[181],[59,17963,17965],{"className":17964},[186,187,188,189],[59,17966,3659],{"className":17967,"style":4157},[103,104,189],"，对每个 ",[59,17970,17972,17989],{"className":17971,"translate":63},[62],[59,17973,17975],{"className":17974},[67],[69,17976,17977],{"xmlns":71},[73,17978,17979,17987],{},[76,17980,17981],{},[132,17982,17983,17985],{},[79,17984,3432],{},[79,17986,3659],{},[83,17988,16145],{"encoding":85},[59,17990,17992],{"className":17991,"ariaHidden":90},[89],[59,17993,17995,17998],{"className":17994},[94],[59,17996],{"className":17997,"style":4680},[98],[59,17999,18001,18004],{"className":18000},[103],[59,18002,3432],{"className":18003},[103,104],[59,18005,18007],{"className":18006},[162],[59,18008,18010],{"className":18009},[166],[59,18011,18013],{"className":18012},[170],[59,18014,18016],{"className":18015,"style":4680},[174],[59,18017,18018,18021],{"style":177},[59,18019],{"className":18020,"style":182},[181],[59,18022,18024],{"className":18023},[186,187,188,189],[59,18025,3659],{"className":18026,"style":4157},[103,104,189]," 找到逼近它的神经元组合 ",[59,18029,18031,18055],{"className":18030,"translate":63},[62],[59,18032,18034],{"className":18033},[67],[69,18035,18036],{"xmlns":71},[73,18037,18038,18052],{},[76,18039,18040,18046,18048,18050],{},[2924,18041,18042,18044],{},[79,18043,15063],{},[79,18045,3659],{},[137,18047,368],{"stretchy":367},[79,18049,3432],{},[137,18051,373],{"stretchy":367},[83,18053,18054],{"encoding":85},"N_k(x)",[59,18056,18058],{"className":18057,"ariaHidden":90},[89],[59,18059,18061,18064,18105,18108,18111],{"className":18060},[94],[59,18062],{"className":18063,"style":386},[98],[59,18065,18067,18070],{"className":18066},[103],[59,18068,15063],{"className":18069,"style":12843},[103,104],[59,18071,18073],{"className":18072},[162],[59,18074,18076,18097],{"className":18075},[166,527],[59,18077,18079,18094],{"className":18078},[170],[59,18080,18082],{"className":18081,"style":8238},[174],[59,18083,18085,18088],{"style":18084},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[59,18086],{"className":18087,"style":182},[181],[59,18089,18091],{"className":18090},[186,187,188,189],[59,18092,3659],{"className":18093,"style":4157},[103,104,189],[59,18095,564],{"className":18096},[563],[59,18098,18100],{"className":18099},[170],[59,18101,18103],{"className":18102,"style":3092},[174],[59,18104],{},[59,18106,368],{"className":18107},[393],[59,18109,3432],{"className":18110},[103,104],[59,18112,373],{"className":18113},[400],"：",[59,18116,18118],{"className":18117,"translate":63},[767],[59,18119,18121,18177],{"className":18120,"translate":63},[62],[59,18122,18124],{"className":18123},[67],[69,18125,18126],{"xmlns":71,"display":776},[73,18127,18128,18174],{},[76,18129,18130,18132,18138,18140,18142,18144,18146,18152,18154,18156],{},[79,18131,6298],{"mathvariant":1091},[2924,18133,18134,18136],{},[79,18135,15063],{},[79,18137,3659],{},[137,18139,368],{"stretchy":367},[79,18141,3432],{},[137,18143,373],{"stretchy":367},[137,18145,6309],{},[132,18147,18148,18150],{},[79,18149,3432],{},[79,18151,3659],{},[79,18153,6298],{"mathvariant":1091},[137,18155,6322],{},[1174,18157,18158,18160],{},[79,18159,15007],{},[76,18161,18162,18164,18166,18168,18170,18172],{},[473,18163,475],{},[137,18165,368],{"stretchy":367},[79,18167,8496],{},[137,18169,2000],{},[473,18171,637],{},[137,18173,373],{"stretchy":367},[83,18175,18176],{"encoding":85},"  |N_k(x) - x^k| \u003C \\frac{\\epsilon}{2 (m+1)}",[59,18178,18180,18247,18294],{"className":18179,"ariaHidden":90},[89],[59,18181,18183,18186,18189,18229,18232,18235,18238,18241,18244],{"className":18182},[94],[59,18184],{"className":18185,"style":386},[98],[59,18187,6298],{"className":18188},[103],[59,18190,18192,18195],{"className":18191},[103],[59,18193,15063],{"className":18194,"style":12843},[103,104],[59,18196,18198],{"className":18197},[162],[59,18199,18201,18221],{"className":18200},[166,527],[59,18202,18204,18218],{"className":18203},[170],[59,18205,18207],{"className":18206,"style":8238},[174],[59,18208,18209,18212],{"style":18084},[59,18210],{"className":18211,"style":182},[181],[59,18213,18215],{"className":18214},[186,187,188,189],[59,18216,3659],{"className":18217,"style":4157},[103,104,189],[59,18219,564],{"className":18220},[563],[59,18222,18224],{"className":18223},[170],[59,18225,18227],{"className":18226,"style":3092},[174],[59,18228],{},[59,18230,368],{"className":18231},[393],[59,18233,3432],{"className":18234},[103,104],[59,18236,373],{"className":18237},[400],[59,18239],{"className":18240,"style":105},[870],[59,18242,6309],{"className":18243},[193],[59,18245],{"className":18246,"style":105},[870],[59,18248,18250,18253,18282,18285,18288,18291],{"className":18249},[94],[59,18251],{"className":18252,"style":6968},[98],[59,18254,18256,18259],{"className":18255},[103],[59,18257,3432],{"className":18258},[103,104],[59,18260,18262],{"className":18261},[162],[59,18263,18265],{"className":18264},[166],[59,18266,18268],{"className":18267},[170],[59,18269,18271],{"className":18270,"style":6935},[174],[59,18272,18273,18276],{"style":1711},[59,18274],{"className":18275,"style":182},[181],[59,18277,18279],{"className":18278},[186,187,188,189],[59,18280,3659],{"className":18281,"style":4157},[103,104,189],[59,18283,6298],{"className":18284},[103],[59,18286],{"className":18287,"style":871},[870],[59,18289,6322],{"className":18290},[875],[59,18292],{"className":18293,"style":871},[870],[59,18295,18297,18301],{"className":18296},[94],[59,18298],{"className":18299,"style":18300},[98],"height:2.0436em;vertical-align:-0.936em;",[59,18302,18304,18307,18382],{"className":18303},[103],[59,18305],{"className":18306},[393,1222],[59,18308,18310],{"className":18309},[1174],[59,18311,18313,18373],{"className":18312},[166,527],[59,18314,18316,18370],{"className":18315},[170],[59,18317,18319,18351,18359],{"className":18318,"style":6877},[174],[59,18320,18321,18324],{"style":4109},[59,18322],{"className":18323,"style":541},[181],[59,18325,18327,18330,18333,18336,18339,18342,18345,18348],{"className":18326},[103],[59,18328,475],{"className":18329},[103],[59,18331,368],{"className":18332},[393],[59,18334,8496],{"className":18335},[103,104],[59,18337],{"className":18338,"style":105},[870],[59,18340,2000],{"className":18341},[193],[59,18343],{"className":18344,"style":105},[870],[59,18346,637],{"className":18347},[103],[59,18349,373],{"className":18350},[400],[59,18352,18353,18356],{"style":1253},[59,18354],{"className":18355,"style":541},[181],[59,18357],{"className":18358,"style":1261},[1260],[59,18360,18361,18364],{"style":4203},[59,18362],{"className":18363,"style":541},[181],[59,18365,18367],{"className":18366},[103],[59,18368,15007],{"className":18369},[103,104],[59,18371,564],{"className":18372},[563],[59,18374,18376],{"className":18375},[170],[59,18377,18380],{"className":18378,"style":18379},[174],"height:0.936em;",[59,18381],{},[59,18383],{"className":18384},[400,1222],[14,18386,18387],{},"然后组合：",[59,18389,18391],{"className":18390,"translate":63},[767],[59,18392,18394,18448],{"className":18393,"translate":63},[62],[59,18395,18397],{"className":18396},[67],[69,18398,18399],{"xmlns":71,"display":776},[73,18400,18401,18445],{},[76,18402,18403,18405,18407,18409,18411,18413,18427,18433,18439,18441,18443],{},[79,18404,15063],{},[137,18406,368],{"stretchy":367},[79,18408,3432],{},[137,18410,373],{"stretchy":367},[137,18412,791],{},[6523,18414,18415,18417,18425],{},[137,18416,6527],{},[76,18418,18419,18421,18423],{},[79,18420,3659],{},[137,18422,791],{},[473,18424,471],{},[79,18426,8496],{},[2924,18428,18429,18431],{},[79,18430,8501],{},[79,18432,3659],{},[2924,18434,18435,18437],{},[79,18436,15063],{},[79,18438,3659],{},[137,18440,368],{"stretchy":367},[79,18442,3432],{},[137,18444,373],{"stretchy":367},[83,18446,18447],{"encoding":85},"  N(x) = \\sum_{k=0}^m a_k N_k(x)",[59,18449,18451,18478],{"className":18450,"ariaHidden":90},[89],[59,18452,18454,18457,18460,18463,18466,18469,18472,18475],{"className":18453},[94],[59,18455],{"className":18456,"style":386},[98],[59,18458,15063],{"className":18459,"style":12843},[103,104],[59,18461,368],{"className":18462},[393],[59,18464,3432],{"className":18465},[103,104],[59,18467,373],{"className":18468},[400],[59,18470],{"className":18471,"style":871},[870],[59,18473,791],{"className":18474},[875],[59,18476],{"className":18477,"style":871},[870],[59,18479,18481,18484,18548,18551,18591,18631,18634,18637],{"className":18480},[94],[59,18482],{"className":18483,"style":6684},[98],[59,18485,18487],{"className":18486},[6688,6689],[59,18488,18490,18540],{"className":18489},[166,527],[59,18491,18493,18537],{"className":18492},[170],[59,18494,18496,18516,18526],{"className":18495,"style":6699},[174],[59,18497,18498,18501],{"style":6702},[59,18499],{"className":18500,"style":6706},[181],[59,18502,18504],{"className":18503},[186,187,188,189],[59,18505,18507,18510,18513],{"className":18506},[103,189],[59,18508,3659],{"className":18509,"style":4157},[103,104,189],[59,18511,791],{"className":18512},[875,189],[59,18514,471],{"className":18515},[103,189],[59,18517,18518,18521],{"style":6724},[59,18519],{"className":18520,"style":6706},[181],[59,18522,18523],{},[59,18524,6527],{"className":18525},[6688,6733,6734],[59,18527,18528,18531],{"style":6737},[59,18529],{"className":18530,"style":6706},[181],[59,18532,18534],{"className":18533},[186,187,188,189],[59,18535,8496],{"className":18536},[103,104,189],[59,18538,564],{"className":18539},[563],[59,18541,18543],{"className":18542},[170],[59,18544,18546],{"className":18545,"style":6759},[174],[59,18547],{},[59,18549],{"className":18550,"style":983},[870],[59,18552,18554,18557],{"className":18553},[103],[59,18555,8501],{"className":18556},[103,104],[59,18558,18560],{"className":18559},[162],[59,18561,18563,18583],{"className":18562},[166,527],[59,18564,18566,18580],{"className":18565},[170],[59,18567,18569],{"className":18568,"style":8238},[174],[59,18570,18571,18574],{"style":8241},[59,18572],{"className":18573,"style":182},[181],[59,18575,18577],{"className":18576},[186,187,188,189],[59,18578,3659],{"className":18579,"style":4157},[103,104,189],[59,18581,564],{"className":18582},[563],[59,18584,18586],{"className":18585},[170],[59,18587,18589],{"className":18588,"style":3092},[174],[59,18590],{},[59,18592,18594,18597],{"className":18593},[103],[59,18595,15063],{"className":18596,"style":12843},[103,104],[59,18598,18600],{"className":18599},[162],[59,18601,18603,18623],{"className":18602},[166,527],[59,18604,18606,18620],{"className":18605},[170],[59,18607,18609],{"className":18608,"style":8238},[174],[59,18610,18611,18614],{"style":18084},[59,18612],{"className":18613,"style":182},[181],[59,18615,18617],{"className":18616},[186,187,188,189],[59,18618,3659],{"className":18619,"style":4157},[103,104,189],[59,18621,564],{"className":18622},[563],[59,18624,18626],{"className":18625},[170],[59,18627,18629],{"className":18628,"style":3092},[174],[59,18630],{},[59,18632,368],{"className":18633},[393],[59,18635,3432],{"className":18636},[103,104],[59,18638,373],{"className":18639},[400],[14,18641,18642],{},"误差界：",[59,18644,18646],{"className":18645,"translate":63},[767],[59,18647,18649,18785],{"className":18648,"translate":63},[62],[59,18650,18652],{"className":18651},[67],[69,18653,18654],{"xmlns":71,"display":776},[73,18655,18656,18782],{},[76,18657,18658,18660,18662,18664,18666,18668,18670,18672,18674,18676,18678,18680,18682,18696,18698,18704,18706,18708,18714,18716,18718,18720,18722,18728,18730,18732,18746,18748,18754,18756,18774,18776,18778,18780],{},[79,18659,6298],{"mathvariant":1091},[79,18661,15063],{},[137,18663,368],{"stretchy":367},[79,18665,3432],{},[137,18667,373],{"stretchy":367},[137,18669,6309],{},[79,18671,6249],{},[137,18673,368],{"stretchy":367},[79,18675,3432],{},[137,18677,373],{"stretchy":367},[79,18679,6298],{"mathvariant":1091},[137,18681,11787],{},[6523,18683,18684,18686,18694],{},[137,18685,6527],{},[76,18687,18688,18690,18692],{},[79,18689,3659],{},[137,18691,791],{},[473,18693,471],{},[79,18695,8496],{},[79,18697,6298],{"mathvariant":1091},[2924,18699,18700,18702],{},[79,18701,8501],{},[79,18703,3659],{},[79,18705,6298],{"mathvariant":1091},[79,18707,6298],{"mathvariant":1091},[2924,18709,18710,18712],{},[79,18711,15063],{},[79,18713,3659],{},[137,18715,368],{"stretchy":367},[79,18717,3432],{},[137,18719,373],{"stretchy":367},[137,18721,6309],{},[132,18723,18724,18726],{},[79,18725,3432],{},[79,18727,3659],{},[79,18729,6298],{"mathvariant":1091},[137,18731,6322],{},[6523,18733,18734,18736,18744],{},[137,18735,6527],{},[76,18737,18738,18740,18742],{},[79,18739,3659],{},[137,18741,791],{},[473,18743,471],{},[79,18745,8496],{},[79,18747,6298],{"mathvariant":1091},[2924,18749,18750,18752],{},[79,18751,8501],{},[79,18753,3659],{},[79,18755,6298],{"mathvariant":1091},[1174,18757,18758,18760],{},[79,18759,15007],{},[76,18761,18762,18764,18766,18768,18770,18772],{},[473,18763,475],{},[137,18765,368],{"stretchy":367},[79,18767,8496],{},[137,18769,2000],{},[473,18771,637],{},[137,18773,373],{"stretchy":367},[137,18775,6322],{},[79,18777,15007],{},[79,18779,1092],{"mathvariant":1091},[473,18781,475],{},[83,18783,18784],{"encoding":85},"  |N(x) - P(x)| \\le \\sum_{k=0}^m |a_k||N_k(x)-x^k| \u003C \\sum_{k=0}^m |a_k| \\frac{\\epsilon}{2(m+1)} \u003C \\epsilon\u002F2",[59,18786,18788,18818,18848,19026,19073,19284],{"className":18787,"ariaHidden":90},[89],[59,18789,18791,18794,18797,18800,18803,18806,18809,18812,18815],{"className":18790},[94],[59,18792],{"className":18793,"style":386},[98],[59,18795,6298],{"className":18796},[103],[59,18798,15063],{"className":18799,"style":12843},[103,104],[59,18801,368],{"className":18802},[393],[59,18804,3432],{"className":18805},[103,104],[59,18807,373],{"className":18808},[400],[59,18810],{"className":18811,"style":105},[870],[59,18813,6309],{"className":18814},[193],[59,18816],{"className":18817,"style":105},[870],[59,18819,18821,18824,18827,18830,18833,18836,18839,18842,18845],{"className":18820},[94],[59,18822],{"className":18823,"style":386},[98],[59,18825,6249],{"className":18826,"style":1412},[103,104],[59,18828,368],{"className":18829},[393],[59,18831,3432],{"className":18832},[103,104],[59,18834,373],{"className":18835},[400],[59,18837,6298],{"className":18838},[103],[59,18840],{"className":18841,"style":871},[870],[59,18843,11787],{"className":18844},[875],[59,18846],{"className":18847,"style":871},[870],[59,18849,18851,18854,18918,18921,18924,18964,18968,19008,19011,19014,19017,19020,19023],{"className":18850},[94],[59,18852],{"className":18853,"style":6684},[98],[59,18855,18857],{"className":18856},[6688,6689],[59,18858,18860,18910],{"className":18859},[166,527],[59,18861,18863,18907],{"className":18862},[170],[59,18864,18866,18886,18896],{"className":18865,"style":6699},[174],[59,18867,18868,18871],{"style":6702},[59,18869],{"className":18870,"style":6706},[181],[59,18872,18874],{"className":18873},[186,187,188,189],[59,18875,18877,18880,18883],{"className":18876},[103,189],[59,18878,3659],{"className":18879,"style":4157},[103,104,189],[59,18881,791],{"className":18882},[875,189],[59,18884,471],{"className":18885},[103,189],[59,18887,18888,18891],{"style":6724},[59,18889],{"className":18890,"style":6706},[181],[59,18892,18893],{},[59,18894,6527],{"className":18895},[6688,6733,6734],[59,18897,18898,18901],{"style":6737},[59,18899],{"className":18900,"style":6706},[181],[59,18902,18904],{"className":18903},[186,187,188,189],[59,18905,8496],{"className":18906},[103,104,189],[59,18908,564],{"className":18909},[563],[59,18911,18913],{"className":18912},[170],[59,18914,18916],{"className":18915,"style":6759},[174],[59,18917],{},[59,18919],{"className":18920,"style":983},[870],[59,18922,6298],{"className":18923},[103],[59,18925,18927,18930],{"className":18926},[103],[59,18928,8501],{"className":18929},[103,104],[59,18931,18933],{"className":18932},[162],[59,18934,18936,18956],{"className":18935},[166,527],[59,18937,18939,18953],{"className":18938},[170],[59,18940,18942],{"className":18941,"style":8238},[174],[59,18943,18944,18947],{"style":8241},[59,18945],{"className":18946,"style":182},[181],[59,18948,18950],{"className":18949},[186,187,188,189],[59,18951,3659],{"className":18952,"style":4157},[103,104,189],[59,18954,564],{"className":18955},[563],[59,18957,18959],{"className":18958},[170],[59,18960,18962],{"className":18961,"style":3092},[174],[59,18963],{},[59,18965,18967],{"className":18966},[103],"∣∣",[59,18969,18971,18974],{"className":18970},[103],[59,18972,15063],{"className":18973,"style":12843},[103,104],[59,18975,18977],{"className":18976},[162],[59,18978,18980,19000],{"className":18979},[166,527],[59,18981,18983,18997],{"className":18982},[170],[59,18984,18986],{"className":18985,"style":8238},[174],[59,18987,18988,18991],{"style":18084},[59,18989],{"className":18990,"style":182},[181],[59,18992,18994],{"className":18993},[186,187,188,189],[59,18995,3659],{"className":18996,"style":4157},[103,104,189],[59,18998,564],{"className":18999},[563],[59,19001,19003],{"className":19002},[170],[59,19004,19006],{"className":19005,"style":3092},[174],[59,19007],{},[59,19009,368],{"className":19010},[393],[59,19012,3432],{"className":19013},[103,104],[59,19015,373],{"className":19016},[400],[59,19018],{"className":19019,"style":105},[870],[59,19021,6309],{"className":19022},[193],[59,19024],{"className":19025,"style":105},[870],[59,19027,19029,19032,19061,19064,19067,19070],{"className":19028},[94],[59,19030],{"className":19031,"style":6968},[98],[59,19033,19035,19038],{"className":19034},[103],[59,19036,3432],{"className":19037},[103,104],[59,19039,19041],{"className":19040},[162],[59,19042,19044],{"className":19043},[166],[59,19045,19047],{"className":19046},[170],[59,19048,19050],{"className":19049,"style":6935},[174],[59,19051,19052,19055],{"style":1711},[59,19053],{"className":19054,"style":182},[181],[59,19056,19058],{"className":19057},[186,187,188,189],[59,19059,3659],{"className":19060,"style":4157},[103,104,189],[59,19062,6298],{"className":19063},[103],[59,19065],{"className":19066,"style":871},[870],[59,19068,6322],{"className":19069},[875],[59,19071],{"className":19072,"style":871},[870],[59,19074,19076,19079,19143,19146,19149,19189,19192,19275,19278,19281],{"className":19075},[94],[59,19077],{"className":19078,"style":6684},[98],[59,19080,19082],{"className":19081},[6688,6689],[59,19083,19085,19135],{"className":19084},[166,527],[59,19086,19088,19132],{"className":19087},[170],[59,19089,19091,19111,19121],{"className":19090,"style":6699},[174],[59,19092,19093,19096],{"style":6702},[59,19094],{"className":19095,"style":6706},[181],[59,19097,19099],{"className":19098},[186,187,188,189],[59,19100,19102,19105,19108],{"className":19101},[103,189],[59,19103,3659],{"className":19104,"style":4157},[103,104,189],[59,19106,791],{"className":19107},[875,189],[59,19109,471],{"className":19110},[103,189],[59,19112,19113,19116],{"style":6724},[59,19114],{"className":19115,"style":6706},[181],[59,19117,19118],{},[59,19119,6527],{"className":19120},[6688,6733,6734],[59,19122,19123,19126],{"style":6737},[59,19124],{"className":19125,"style":6706},[181],[59,19127,19129],{"className":19128},[186,187,188,189],[59,19130,8496],{"className":19131},[103,104,189],[59,19133,564],{"className":19134},[563],[59,19136,19138],{"className":19137},[170],[59,19139,19141],{"className":19140,"style":6759},[174],[59,19142],{},[59,19144],{"className":19145,"style":983},[870],[59,19147,6298],{"className":19148},[103],[59,19150,19152,19155],{"className":19151},[103],[59,19153,8501],{"className":19154},[103,104],[59,19156,19158],{"className":19157},[162],[59,19159,19161,19181],{"className":19160},[166,527],[59,19162,19164,19178],{"className":19163},[170],[59,19165,19167],{"className":19166,"style":8238},[174],[59,19168,19169,19172],{"style":8241},[59,19170],{"className":19171,"style":182},[181],[59,19173,19175],{"className":19174},[186,187,188,189],[59,19176,3659],{"className":19177,"style":4157},[103,104,189],[59,19179,564],{"className":19180},[563],[59,19182,19184],{"className":19183},[170],[59,19185,19187],{"className":19186,"style":3092},[174],[59,19188],{},[59,19190,6298],{"className":19191},[103],[59,19193,19195,19198,19272],{"className":19194},[103],[59,19196],{"className":19197},[393,1222],[59,19199,19201],{"className":19200},[1174],[59,19202,19204,19264],{"className":19203},[166,527],[59,19205,19207,19261],{"className":19206},[170],[59,19208,19210,19242,19250],{"className":19209,"style":6877},[174],[59,19211,19212,19215],{"style":4109},[59,19213],{"className":19214,"style":541},[181],[59,19216,19218,19221,19224,19227,19230,19233,19236,19239],{"className":19217},[103],[59,19219,475],{"className":19220},[103],[59,19222,368],{"className":19223},[393],[59,19225,8496],{"className":19226},[103,104],[59,19228],{"className":19229,"style":105},[870],[59,19231,2000],{"className":19232},[193],[59,19234],{"className":19235,"style":105},[870],[59,19237,637],{"className":19238},[103],[59,19240,373],{"className":19241},[400],[59,19243,19244,19247],{"style":1253},[59,19245],{"className":19246,"style":541},[181],[59,19248],{"className":19249,"style":1261},[1260],[59,19251,19252,19255],{"style":4203},[59,19253],{"className":19254,"style":541},[181],[59,19256,19258],{"className":19257},[103],[59,19259,15007],{"className":19260},[103,104],[59,19262,564],{"className":19263},[563],[59,19265,19267],{"className":19266},[170],[59,19268,19270],{"className":19269,"style":18379},[174],[59,19271],{},[59,19273],{"className":19274},[400,1222],[59,19276],{"className":19277,"style":871},[870],[59,19279,6322],{"className":19280},[875],[59,19282],{"className":19283,"style":871},[870],[59,19285,19287,19290,19293],{"className":19286},[94],[59,19288],{"className":19289,"style":386},[98],[59,19291,15007],{"className":19292},[103,104],[59,19294,15855],{"className":19295},[103],[14,19297,19298],{},"第 4 步：三角不等式完成逼近",[14,19300,19301],{},"我们有两处误差来源：",[19303,19304,19305,19428],"ol",{},[409,19306,19307,19427],{},[59,19308,19310,19352],{"className":19309,"translate":63},[62],[59,19311,19313],{"className":19312},[67],[69,19314,19315],{"xmlns":71},[73,19316,19317,19349],{},[76,19318,19319,19321,19323,19325,19327,19329,19331,19333,19335,19337,19339,19341,19343,19345,19347],{},[79,19320,6298],{"mathvariant":1091},[79,19322,6010],{},[137,19324,368],{"stretchy":367},[79,19326,3432],{},[137,19328,373],{"stretchy":367},[137,19330,6309],{},[79,19332,6249],{},[137,19334,368],{"stretchy":367},[79,19336,3432],{},[137,19338,373],{"stretchy":367},[79,19340,6298],{"mathvariant":1091},[137,19342,6322],{},[79,19344,15007],{},[79,19346,1092],{"mathvariant":1091},[473,19348,475],{},[83,19350,19351],{"encoding":85},"|f(x) - P(x)| \u003C \\epsilon\u002F2",[59,19353,19355,19385,19415],{"className":19354,"ariaHidden":90},[89],[59,19356,19358,19361,19364,19367,19370,19373,19376,19379,19382],{"className":19357},[94],[59,19359],{"className":19360,"style":386},[98],[59,19362,6298],{"className":19363},[103],[59,19365,6010],{"className":19366,"style":6047},[103,104],[59,19368,368],{"className":19369},[393],[59,19371,3432],{"className":19372},[103,104],[59,19374,373],{"className":19375},[400],[59,19377],{"className":19378,"style":105},[870],[59,19380,6309],{"className":19381},[193],[59,19383],{"className":19384,"style":105},[870],[59,19386,19388,19391,19394,19397,19400,19403,19406,19409,19412],{"className":19387},[94],[59,19389],{"className":19390,"style":386},[98],[59,19392,6249],{"className":19393,"style":1412},[103,104],[59,19395,368],{"className":19396},[393],[59,19398,3432],{"className":19399},[103,104],[59,19401,373],{"className":19402},[400],[59,19404,6298],{"className":19405},[103],[59,19407],{"className":19408,"style":871},[870],[59,19410,6322],{"className":19411},[875],[59,19413],{"className":19414,"style":871},[870],[59,19416,19418,19421,19424],{"className":19417},[94],[59,19419],{"className":19420,"style":386},[98],[59,19422,15007],{"className":19423},[103,104],[59,19425,15855],{"className":19426},[103],"（魏尔斯特拉斯定理）",[409,19429,19430,19550],{},[59,19431,19433,19475],{"className":19432,"translate":63},[62],[59,19434,19436],{"className":19435},[67],[69,19437,19438],{"xmlns":71},[73,19439,19440,19472],{},[76,19441,19442,19444,19446,19448,19450,19452,19454,19456,19458,19460,19462,19464,19466,19468,19470],{},[79,19443,6298],{"mathvariant":1091},[79,19445,6249],{},[137,19447,368],{"stretchy":367},[79,19449,3432],{},[137,19451,373],{"stretchy":367},[137,19453,6309],{},[79,19455,15063],{},[137,19457,368],{"stretchy":367},[79,19459,3432],{},[137,19461,373],{"stretchy":367},[79,19463,6298],{"mathvariant":1091},[137,19465,6322],{},[79,19467,15007],{},[79,19469,1092],{"mathvariant":1091},[473,19471,475],{},[83,19473,19474],{"encoding":85},"|P(x) - N(x)| \u003C \\epsilon\u002F2",[59,19476,19478,19508,19538],{"className":19477,"ariaHidden":90},[89],[59,19479,19481,19484,19487,19490,19493,19496,19499,19502,19505],{"className":19480},[94],[59,19482],{"className":19483,"style":386},[98],[59,19485,6298],{"className":19486},[103],[59,19488,6249],{"className":19489,"style":1412},[103,104],[59,19491,368],{"className":19492},[393],[59,19494,3432],{"className":19495},[103,104],[59,19497,373],{"className":19498},[400],[59,19500],{"className":19501,"style":105},[870],[59,19503,6309],{"className":19504},[193],[59,19506],{"className":19507,"style":105},[870],[59,19509,19511,19514,19517,19520,19523,19526,19529,19532,19535],{"className":19510},[94],[59,19512],{"className":19513,"style":386},[98],[59,19515,15063],{"className":19516,"style":12843},[103,104],[59,19518,368],{"className":19519},[393],[59,19521,3432],{"className":19522},[103,104],[59,19524,373],{"className":19525},[400],[59,19527,6298],{"className":19528},[103],[59,19530],{"className":19531,"style":871},[870],[59,19533,6322],{"className":19534},[875],[59,19536],{"className":19537,"style":871},[870],[59,19539,19541,19544,19547],{"className":19540},[94],[59,19542],{"className":19543,"style":386},[98],[59,19545,15007],{"className":19546},[103,104],[59,19548,15855],{"className":19549},[103],"（神经元对多项式的逼近）",[14,19552,19553],{},"由三角不等式：",[59,19555,19557],{"className":19556,"translate":63},[767],[59,19558,19560,19662],{"className":19559,"translate":63},[62],[59,19561,19563],{"className":19562},[67],[69,19564,19565],{"xmlns":71,"display":776},[73,19566,19567,19659],{},[76,19568,19569,19571,19573,19575,19577,19579,19581,19583,19585,19587,19589,19591,19593,19595,19597,19599,19601,19603,19605,19607,19609,19611,19613,19615,19617,19619,19621,19623,19625,19627,19629,19631,19633,19635,19637,19639,19641,19643,19645,19647,19649,19651,19653,19655,19657],{},[79,19570,6298],{"mathvariant":1091},[79,19572,6010],{},[137,19574,368],{"stretchy":367},[79,19576,3432],{},[137,19578,373],{"stretchy":367},[137,19580,6309],{},[79,19582,15063],{},[137,19584,368],{"stretchy":367},[79,19586,3432],{},[137,19588,373],{"stretchy":367},[79,19590,6298],{"mathvariant":1091},[137,19592,11787],{},[79,19594,6298],{"mathvariant":1091},[79,19596,6010],{},[137,19598,368],{"stretchy":367},[79,19600,3432],{},[137,19602,373],{"stretchy":367},[137,19604,6309],{},[79,19606,6249],{},[137,19608,368],{"stretchy":367},[79,19610,3432],{},[137,19612,373],{"stretchy":367},[79,19614,6298],{"mathvariant":1091},[137,19616,2000],{},[79,19618,6298],{"mathvariant":1091},[79,19620,6249],{},[137,19622,368],{"stretchy":367},[79,19624,3432],{},[137,19626,373],{"stretchy":367},[137,19628,6309],{},[79,19630,15063],{},[137,19632,368],{"stretchy":367},[79,19634,3432],{},[137,19636,373],{"stretchy":367},[79,19638,6298],{"mathvariant":1091},[137,19640,6322],{},[79,19642,15007],{},[79,19644,1092],{"mathvariant":1091},[473,19646,475],{},[137,19648,2000],{},[79,19650,15007],{},[79,19652,1092],{"mathvariant":1091},[473,19654,475],{},[137,19656,791],{},[79,19658,15007],{},[83,19660,19661],{"encoding":85},"  |f(x) - N(x)| \\le |f(x)-P(x)| + |P(x)-N(x)| \u003C \\epsilon\u002F2 + \\epsilon\u002F2 = \\epsilon",[59,19663,19665,19695,19725,19755,19785,19815,19845,19866,19887],{"className":19664,"ariaHidden":90},[89],[59,19666,19668,19671,19674,19677,19680,19683,19686,19689,19692],{"className":19667},[94],[59,19669],{"className":19670,"style":386},[98],[59,19672,6298],{"className":19673},[103],[59,19675,6010],{"className":19676,"style":6047},[103,104],[59,19678,368],{"className":19679},[393],[59,19681,3432],{"className":19682},[103,104],[59,19684,373],{"className":19685},[400],[59,19687],{"className":19688,"style":105},[870],[59,19690,6309],{"className":19691},[193],[59,19693],{"className":19694,"style":105},[870],[59,19696,19698,19701,19704,19707,19710,19713,19716,19719,19722],{"className":19697},[94],[59,19699],{"className":19700,"style":386},[98],[59,19702,15063],{"className":19703,"style":12843},[103,104],[59,19705,368],{"className":19706},[393],[59,19708,3432],{"className":19709},[103,104],[59,19711,373],{"className":19712},[400],[59,19714,6298],{"className":19715},[103],[59,19717],{"className":19718,"style":871},[870],[59,19720,11787],{"className":19721},[875],[59,19723],{"className":19724,"style":871},[870],[59,19726,19728,19731,19734,19737,19740,19743,19746,19749,19752],{"className":19727},[94],[59,19729],{"className":19730,"style":386},[98],[59,19732,6298],{"className":19733},[103],[59,19735,6010],{"className":19736,"style":6047},[103,104],[59,19738,368],{"className":19739},[393],[59,19741,3432],{"className":19742},[103,104],[59,19744,373],{"className":19745},[400],[59,19747],{"className":19748,"style":105},[870],[59,19750,6309],{"className":19751},[193],[59,19753],{"className":19754,"style":105},[870],[59,19756,19758,19761,19764,19767,19770,19773,19776,19779,19782],{"className":19757},[94],[59,19759],{"className":19760,"style":386},[98],[59,19762,6249],{"className":19763,"style":1412},[103,104],[59,19765,368],{"className":19766},[393],[59,19768,3432],{"className":19769},[103,104],[59,19771,373],{"className":19772},[400],[59,19774,6298],{"className":19775},[103],[59,19777],{"className":19778,"style":105},[870],[59,19780,2000],{"className":19781},[193],[59,19783],{"className":19784,"style":105},[870],[59,19786,19788,19791,19794,19797,19800,19803,19806,19809,19812],{"className":19787},[94],[59,19789],{"className":19790,"style":386},[98],[59,19792,6298],{"className":19793},[103],[59,19795,6249],{"className":19796,"style":1412},[103,104],[59,19798,368],{"className":19799},[393],[59,19801,3432],{"className":19802},[103,104],[59,19804,373],{"className":19805},[400],[59,19807],{"className":19808,"style":105},[870],[59,19810,6309],{"className":19811},[193],[59,19813],{"className":19814,"style":105},[870],[59,19816,19818,19821,19824,19827,19830,19833,19836,19839,19842],{"className":19817},[94],[59,19819],{"className":19820,"style":386},[98],[59,19822,15063],{"className":19823,"style":12843},[103,104],[59,19825,368],{"className":19826},[393],[59,19828,3432],{"className":19829},[103,104],[59,19831,373],{"className":19832},[400],[59,19834,6298],{"className":19835},[103],[59,19837],{"className":19838,"style":871},[870],[59,19840,6322],{"className":19841},[875],[59,19843],{"className":19844,"style":871},[870],[59,19846,19848,19851,19854,19857,19860,19863],{"className":19847},[94],[59,19849],{"className":19850,"style":386},[98],[59,19852,15007],{"className":19853},[103,104],[59,19855,15855],{"className":19856},[103],[59,19858],{"className":19859,"style":105},[870],[59,19861,2000],{"className":19862},[193],[59,19864],{"className":19865,"style":105},[870],[59,19867,19869,19872,19875,19878,19881,19884],{"className":19868},[94],[59,19870],{"className":19871,"style":386},[98],[59,19873,15007],{"className":19874},[103,104],[59,19876,15855],{"className":19877},[103],[59,19879],{"className":19880,"style":871},[870],[59,19882,791],{"className":19883},[875],[59,19885],{"className":19886,"style":871},[870],[59,19888,19890,19893],{"className":19889},[94],[59,19891],{"className":19892,"style":310},[98],[59,19894,15007],{"className":19895},[103,104],[14,19897,19898],{},"证毕。",[14,19900,19901],{},"换言之：神经网络的非线性单元充当「可调节的多项式基」，单层就能把它们组合成任意多项式，从而逼近任何连续函数。",[30,19903,19905],{"id":19904},"uat-与机器学习视角","UAT 与机器学习视角",[14,19907,19908],{},"这个证明其实相当有意思：",[14,19910,19911],{},"伯恩斯坦多项式本质上是在表达",[59,19913,19915],{"className":19914,"translate":63},[767],[59,19916,19918,20013],{"className":19917,"translate":63},[62],[59,19919,19921],{"className":19920},[67],[69,19922,19923],{"xmlns":71,"display":776},[73,19924,19925,20010],{},[76,19926,19927,19929,19931,19933,19935,19937,19943,19945,19947,19959,19961],{},[79,19928,6010],{},[137,19930,368],{"stretchy":367},[79,19932,3432],{},[137,19934,373],{"stretchy":367},[137,19936,10405],{},[1497,19938,19939,19941],{},[137,19940,6527],{},[79,19942,3659],{},[79,19944,6010],{},[1991,19946,6542],{},[76,19948,19949,19951,19957],{},[137,19950,368],{"fence":90},[1174,19952,19953,19955],{},[79,19954,3659],{},[79,19956,2119],{},[137,19958,373],{"fence":90},[137,19960,2316],{},[1497,19962,19963,20007],{},[1497,19964,19965,20005],{},[76,19966,19967,19979,19985,19987,19989,19991,19993],{},[76,19968,19969,19971,19977],{},[137,19970,368],{"fence":90},[1174,19972,19973,19975],{"linethickness":6561},[79,19974,2119],{},[79,19976,3659],{},[137,19978,373],{"fence":90},[132,19980,19981,19983],{},[79,19982,3432],{},[79,19984,3659],{},[137,19986,368],{"stretchy":367},[473,19988,637],{},[137,19990,6309],{},[79,19992,3432],{},[132,19994,19995,19997],{},[137,19996,373],{"stretchy":367},[76,19998,19999,20001,20003],{},[79,20000,2119],{},[137,20002,6309],{},[79,20004,3659],{},[137,20006,1524],{"stretchy":90},[1991,20008,20009],{},"基函数",[83,20011,20012],{"encoding":85},"f(x)\n\\approx\n\\sum_k\nf\\!\\left(\\frac{k}{n}\\right)\n\\cdot\n\\underbrace{\n\\binom{n}{k}\nx^k(1-x)^{n-k}\n}_{\\text{基函数}}",[59,20014,20016,20043,20192],{"className":20015,"ariaHidden":90},[89],[59,20017,20019,20022,20025,20028,20031,20034,20037,20040],{"className":20018},[94],[59,20020],{"className":20021,"style":386},[98],[59,20023,6010],{"className":20024,"style":6047},[103,104],[59,20026,368],{"className":20027},[393],[59,20029,3432],{"className":20030},[103,104],[59,20032,373],{"className":20033},[400],[59,20035],{"className":20036,"style":871},[870],[59,20038,10405],{"className":20039},[875],[59,20041],{"className":20042,"style":871},[870],[59,20044,20046,20050,20094,20097,20100,20103,20106,20183,20186,20189],{"className":20045},[94],[59,20047],{"className":20048,"style":20049},[98],"height:2.7521em;vertical-align:-1.3021em;",[59,20051,20053],{"className":20052},[6688,6689],[59,20054,20056,20086],{"className":20055},[166,527],[59,20057,20059,20083],{"className":20058},[170],[59,20060,20062,20073],{"className":20061,"style":17486},[174],[59,20063,20064,20067],{"style":6702},[59,20065],{"className":20066,"style":6706},[181],[59,20068,20070],{"className":20069},[186,187,188,189],[59,20071,3659],{"className":20072,"style":4157},[103,104,189],[59,20074,20075,20078],{"style":6724},[59,20076],{"className":20077,"style":6706},[181],[59,20079,20080],{},[59,20081,6527],{"className":20082},[6688,6733,6734],[59,20084,564],{"className":20085},[563],[59,20087,20089],{"className":20088},[170],[59,20090,20092],{"className":20091,"style":6759},[174],[59,20093],{},[59,20095],{"className":20096,"style":983},[870],[59,20098,6010],{"className":20099,"style":6047},[103,104],[59,20101],{"className":20102,"style":6771},[870],[59,20104],{"className":20105,"style":983},[870],[59,20107,20109,20115,20177],{"className":20108},[504],[59,20110,20112],{"className":20111,"style":509},[393,508],[59,20113,368],{"className":20114},[513,188],[59,20116,20118,20121,20174],{"className":20117},[103],[59,20119],{"className":20120},[393,1222],[59,20122,20124],{"className":20123},[1174],[59,20125,20127,20166],{"className":20126},[166,527],[59,20128,20130,20163],{"className":20129},[170],[59,20131,20133,20144,20152],{"className":20132,"style":6802},[174],[59,20134,20135,20138],{"style":4109},[59,20136],{"className":20137,"style":541},[181],[59,20139,20141],{"className":20140},[103],[59,20142,2119],{"className":20143},[103,104],[59,20145,20146,20149],{"style":1253},[59,20147],{"className":20148,"style":541},[181],[59,20150],{"className":20151,"style":1261},[1260],[59,20153,20154,20157],{"style":4203},[59,20155],{"className":20156,"style":541},[181],[59,20158,20160],{"className":20159},[103],[59,20161,3659],{"className":20162,"style":4157},[103,104],[59,20164,564],{"className":20165},[563],[59,20167,20169],{"className":20168},[170],[59,20170,20172],{"className":20171,"style":4298},[174],[59,20173],{},[59,20175],{"className":20176},[400,1222],[59,20178,20180],{"className":20179,"style":509},[400,508],[59,20181,373],{"className":20182},[513,188],[59,20184],{"className":20185,"style":105},[870],[59,20187,2316],{"className":20188},[193],[59,20190],{"className":20191,"style":105},[870],[59,20193,20195,20199],{"className":20194},[94],[59,20196],{"className":20197,"style":20198},[98],"height:3.7264em;vertical-align:-2.2764em;",[59,20200,20202],{"className":20201},[504,1497],[59,20203,20205,20447],{"className":20204},[166,527],[59,20206,20208,20444],{"className":20207},[170],[59,20209,20211,20230],{"className":20210,"style":534},[174],[59,20212,20214,20218],{"style":20213},"top:-1.1736em;",[59,20215],{"className":20216,"style":20217},[181],"height:3.45em;",[59,20219,20221],{"className":20220},[186,187,188,189],[59,20222,20224],{"className":20223},[103,189],[59,20225,20227],{"className":20226},[103,2018,189],[59,20228,20009],{"className":20229},[103,2022,189],[59,20231,20233,20236],{"style":20232},"top:-3.45em;",[59,20234],{"className":20235,"style":20217},[181],[59,20237,20239],{"className":20238},[504,1497],[59,20240,20242,20435],{"className":20241},[166,527],[59,20243,20245,20432],{"className":20244},[170],[59,20246,20248,20279],{"className":20247,"style":534},[174],[59,20249,20252,20255],{"className":20250,"style":20251},[1636],"top:-1.852em;",[59,20253],{"className":20254,"style":20217},[181],[59,20256,20258,20265,20272],{"className":20257,"style":1645},[1644],[59,20259,20261],{"className":20260,"style":1650},[1649],[1652,20262,20263],{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1658},[1660,20264],{"d":1662},[59,20266,20268],{"className":20267,"style":1650},[1666],[1652,20269,20270],{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1669},[1660,20271],{"d":1672},[59,20273,20275],{"className":20274,"style":1650},[1676],[1652,20276,20277],{"xmlns":1654,"width":1655,"height":1656,"viewBox":1657,"preserveAspectRatio":1679},[1660,20278],{"d":1682},[59,20280,20281,20284],{"style":20232},[59,20282],{"className":20283,"style":20217},[181],[59,20285,20287,20347,20376,20379,20382,20385,20388,20391,20394],{"className":20286},[103],[59,20288,20290,20296,20341],{"className":20289},[103],[59,20291,20293],{"className":20292,"style":509},[393,508],[59,20294,368],{"className":20295},[513,188],[59,20297,20299],{"className":20298},[1174],[59,20300,20302,20333],{"className":20301},[166,527],[59,20303,20305,20330],{"className":20304},[170],[59,20306,20308,20319],{"className":20307,"style":6877},[174],[59,20309,20310,20313],{"style":4109},[59,20311],{"className":20312,"style":541},[181],[59,20314,20316],{"className":20315},[103],[59,20317,3659],{"className":20318,"style":4157},[103,104],[59,20320,20321,20324],{"style":4203},[59,20322],{"className":20323,"style":541},[181],[59,20325,20327],{"className":20326},[103],[59,20328,2119],{"className":20329},[103,104],[59,20331,564],{"className":20332},[563],[59,20334,20336],{"className":20335},[170],[59,20337,20339],{"className":20338,"style":4298},[174],[59,20340],{},[59,20342,20344],{"className":20343,"style":509},[400,508],[59,20345,373],{"className":20346},[513,188],[59,20348,20350,20353],{"className":20349},[103],[59,20351,3432],{"className":20352},[103,104],[59,20354,20356],{"className":20355},[162],[59,20357,20359],{"className":20358},[166],[59,20360,20362],{"className":20361},[170],[59,20363,20365],{"className":20364,"style":6935},[174],[59,20366,20367,20370],{"style":1711},[59,20368],{"className":20369,"style":182},[181],[59,20371,20373],{"className":20372},[186,187,188,189],[59,20374,3659],{"className":20375,"style":4157},[103,104,189],[59,20377,368],{"className":20378},[393],[59,20380,637],{"className":20381},[103],[59,20383],{"className":20384,"style":105},[870],[59,20386,6309],{"className":20387},[193],[59,20389],{"className":20390,"style":105},[870],[59,20392,3432],{"className":20393},[103,104],[59,20395,20397,20400],{"className":20396},[400],[59,20398,373],{"className":20399},[400],[59,20401,20403],{"className":20402},[162],[59,20404,20406],{"className":20405},[166],[59,20407,20409],{"className":20408},[170],[59,20410,20412],{"className":20411,"style":6935},[174],[59,20413,20414,20417],{"style":1711},[59,20415],{"className":20416,"style":182},[181],[59,20418,20420],{"className":20419},[186,187,188,189],[59,20421,20423,20426,20429],{"className":20422},[103,189],[59,20424,2119],{"className":20425},[103,104,189],[59,20427,6309],{"className":20428},[193,189],[59,20430,3659],{"className":20431,"style":4157},[103,104,189],[59,20433,564],{"className":20434},[563],[59,20436,20438],{"className":20437},[170],[59,20439,20442],{"className":20440,"style":20441},[174],"height:1.598em;",[59,20443],{},[59,20445,564],{"className":20446},[563],[59,20448,20450],{"className":20449},[170],[59,20451,20454],{"className":20452,"style":20453},[174],"height:2.2764em;",[59,20455],{},[14,20457,20458],{},"这与神经网络",[59,20460,20462],{"className":20461,"translate":63},[767],[59,20463,20465,20524],{"className":20464,"translate":63},[62],[59,20466,20468],{"className":20467},[67],[69,20469,20470],{"xmlns":71,"display":776},[73,20471,20472,20521],{},[76,20473,20474,20476,20478,20480,20482,20484,20490,20496,20498,20500,20502,20509,20511,20513,20519],{},[79,20475,6010],{},[137,20477,368],{"stretchy":367},[79,20479,3432],{},[137,20481,373],{"stretchy":367},[137,20483,10405],{},[1497,20485,20486,20488],{},[137,20487,6527],{},[79,20489,2933],{},[2924,20491,20492,20494],{},[79,20493,8501],{},[79,20495,2933],{},[137,20497,2316],{"separator":90},[79,20499,2088],{},[137,20501,368],{"stretchy":367},[2924,20503,20504,20507],{},[79,20505,20506],{},"w",[79,20508,2933],{},[79,20510,3432],{},[137,20512,2000],{},[2924,20514,20515,20517],{},[79,20516,14919],{},[79,20518,2933],{},[137,20520,373],{"stretchy":367},[83,20522,20523],{"encoding":85},"f(x)\n\\approx\n\\sum_i\na_i·\\sigma(w_i x+b_i)",[59,20525,20527,20554,20713],{"className":20526,"ariaHidden":90},[89],[59,20528,20530,20533,20536,20539,20542,20545,20548,20551],{"className":20529},[94],[59,20531],{"className":20532,"style":386},[98],[59,20534,6010],{"className":20535,"style":6047},[103,104],[59,20537,368],{"className":20538},[393],[59,20540,3432],{"className":20541},[103,104],[59,20543,373],{"className":20544},[400],[59,20546],{"className":20547,"style":871},[870],[59,20549,10405],{"className":20550},[875],[59,20552],{"className":20553,"style":871},[870],[59,20555,20557,20560,20604,20607,20647,20650,20653,20656,20659,20701,20704,20707,20710],{"className":20556},[94],[59,20558],{"className":20559,"style":17473},[98],[59,20561,20563],{"className":20562},[6688,6689],[59,20564,20566,20596],{"className":20565},[166,527],[59,20567,20569,20593],{"className":20568},[170],[59,20570,20572,20583],{"className":20571,"style":17486},[174],[59,20573,20574,20577],{"style":15171},[59,20575],{"className":20576,"style":6706},[181],[59,20578,20580],{"className":20579},[186,187,188,189],[59,20581,2933],{"className":20582},[103,104,189],[59,20584,20585,20588],{"style":6724},[59,20586],{"className":20587,"style":6706},[181],[59,20589,20590],{},[59,20591,6527],{"className":20592},[6688,6733,6734],[59,20594,564],{"className":20595},[563],[59,20597,20599],{"className":20598},[170],[59,20600,20602],{"className":20601,"style":15223},[174],[59,20603],{},[59,20605],{"className":20606,"style":983},[870],[59,20608,20610,20613],{"className":20609},[103],[59,20611,8501],{"className":20612},[103,104],[59,20614,20616],{"className":20615},[162],[59,20617,20619,20639],{"className":20618},[166,527],[59,20620,20622,20636],{"className":20621},[170],[59,20623,20625],{"className":20624,"style":2968},[174],[59,20626,20627,20630],{"style":8241},[59,20628],{"className":20629,"style":182},[181],[59,20631,20633],{"className":20632},[186,187,188,189],[59,20634,2933],{"className":20635},[103,104,189],[59,20637,564],{"className":20638},[563],[59,20640,20642],{"className":20641},[170],[59,20643,20645],{"className":20644,"style":3092},[174],[59,20646],{},[59,20648,2316],{"className":20649},[1372],[59,20651],{"className":20652,"style":983},[870],[59,20654,2088],{"className":20655,"style":314},[103,104],[59,20657,368],{"className":20658},[393],[59,20660,20662,20666],{"className":20661},[103],[59,20663,20506],{"className":20664,"style":20665},[103,104],"margin-right:0.0269em;",[59,20667,20669],{"className":20668},[162],[59,20670,20672,20693],{"className":20671},[166,527],[59,20673,20675,20690],{"className":20674},[170],[59,20676,20678],{"className":20677,"style":2968},[174],[59,20679,20681,20684],{"style":20680},"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;",[59,20682],{"className":20683,"style":182},[181],[59,20685,20687],{"className":20686},[186,187,188,189],[59,20688,2933],{"className":20689},[103,104,189],[59,20691,564],{"className":20692},[563],[59,20694,20696],{"className":20695},[170],[59,20697,20699],{"className":20698,"style":3092},[174],[59,20700],{},[59,20702,3432],{"className":20703},[103,104],[59,20705],{"className":20706,"style":105},[870],[59,20708,2000],{"className":20709},[193],[59,20711],{"className":20712,"style":105},[870],[59,20714,20716,20719,20759],{"className":20715},[94],[59,20717],{"className":20718,"style":386},[98],[59,20720,20722,20725],{"className":20721},[103],[59,20723,14919],{"className":20724},[103,104],[59,20726,20728],{"className":20727},[162],[59,20729,20731,20751],{"className":20730},[166,527],[59,20732,20734,20748],{"className":20733},[170],[59,20735,20737],{"className":20736,"style":2968},[174],[59,20738,20739,20742],{"style":8241},[59,20740],{"className":20741,"style":182},[181],[59,20743,20745],{"className":20744},[186,187,188,189],[59,20746,2933],{"className":20747},[103,104,189],[59,20749,564],{"className":20750},[563],[59,20752,20754],{"className":20753},[170],[59,20755,20757],{"className":20756,"style":3092},[174],[59,20758],{},[59,20760,373],{"className":20761},[400],[14,20763,20764],{},"非常相似。",[14,20766,20767],{},"差别仅在于：",[406,20769,20770,20915],{},[409,20771,20772,20773],{},"魏尔斯特拉斯使用多项式基 ",[59,20774,20776,20814],{"className":20775,"translate":63},[62],[59,20777,20779],{"className":20778},[67],[69,20780,20781],{"xmlns":71},[73,20782,20783,20811],{},[76,20784,20785,20791,20793,20795,20797,20799],{},[132,20786,20787,20789],{},[79,20788,3432],{},[79,20790,3659],{},[137,20792,368],{"stretchy":367},[473,20794,637],{},[137,20796,6309],{},[79,20798,3432],{},[132,20800,20801,20803],{},[137,20802,373],{"stretchy":367},[76,20804,20805,20807,20809],{},[79,20806,2119],{},[137,20808,6309],{},[79,20810,3659],{},[83,20812,20813],{"encoding":85},"x^k(1-x)^{n-k}",[59,20815,20817,20868],{"className":20816,"ariaHidden":90},[89],[59,20818,20820,20824,20853,20856,20859,20862,20865],{"className":20819},[94],[59,20821],{"className":20822,"style":20823},[98],"height:1.0991em;vertical-align:-0.25em;",[59,20825,20827,20830],{"className":20826},[103],[59,20828,3432],{"className":20829},[103,104],[59,20831,20833],{"className":20832},[162],[59,20834,20836],{"className":20835},[166],[59,20837,20839],{"className":20838},[170],[59,20840,20842],{"className":20841,"style":4680},[174],[59,20843,20844,20847],{"style":177},[59,20845],{"className":20846,"style":182},[181],[59,20848,20850],{"className":20849},[186,187,188,189],[59,20851,3659],{"className":20852,"style":4157},[103,104,189],[59,20854,368],{"className":20855},[393],[59,20857,637],{"className":20858},[103],[59,20860],{"className":20861,"style":105},[870],[59,20863,6309],{"className":20864},[193],[59,20866],{"className":20867,"style":105},[870],[59,20869,20871,20874,20877],{"className":20870},[94],[59,20872],{"className":20873,"style":20823},[98],[59,20875,3432],{"className":20876},[103,104],[59,20878,20880,20883],{"className":20879},[400],[59,20881,373],{"className":20882},[400],[59,20884,20886],{"className":20885},[162],[59,20887,20889],{"className":20888},[166],[59,20890,20892],{"className":20891},[170],[59,20893,20895],{"className":20894,"style":4680},[174],[59,20896,20897,20900],{"style":177},[59,20898],{"className":20899,"style":182},[181],[59,20901,20903],{"className":20902},[186,187,188,189],[59,20904,20906,20909,20912],{"className":20905},[103,189],[59,20907,2119],{"className":20908},[103,104,189],[59,20910,6309],{"className":20911},[193,189],[59,20913,3659],{"className":20914,"style":4157},[103,104,189],[409,20916,20917,20918],{},"神经网络使用激活函数基 ",[59,20919,20921,20947],{"className":20920,"translate":63},[62],[59,20922,20924],{"className":20923},[67],[69,20925,20926],{"xmlns":71},[73,20927,20928,20944],{},[76,20929,20930,20932,20934,20936,20938,20940,20942],{},[79,20931,2088],{},[137,20933,368],{"stretchy":367},[79,20935,20506],{},[79,20937,3432],{},[137,20939,2000],{},[79,20941,14919],{},[137,20943,373],{"stretchy":367},[83,20945,20946],{"encoding":85},"\\sigma(wx+b)",[59,20948,20950,20977],{"className":20949,"ariaHidden":90},[89],[59,20951,20953,20956,20959,20962,20965,20968,20971,20974],{"className":20952},[94],[59,20954],{"className":20955,"style":386},[98],[59,20957,2088],{"className":20958,"style":314},[103,104],[59,20960,368],{"className":20961},[393],[59,20963,20506],{"className":20964,"style":20665},[103,104],[59,20966,3432],{"className":20967},[103,104],[59,20969],{"className":20970,"style":105},[870],[59,20972,2000],{"className":20973},[193],[59,20975],{"className":20976,"style":105},[870],[59,20978,20980,20983,20986],{"className":20979},[94],[59,20981],{"className":20982,"style":386},[98],[59,20984,14919],{"className":20985},[103,104],[59,20987,373],{"className":20988},[400],[14,20990,20991],{},"因此，从现代视角看，UAT 定理可以理解为魏尔斯特拉斯定理向「神经网络基函数」的推广。魏尔斯特拉斯说「多项式基是稠密的」，UAT 则说「神经元基也是稠密的」。",{"title":5952,"searchDepth":5953,"depth":5953,"links":20993},[20994,20999,21000],{"id":5985,"depth":5953,"text":5985,"children":20995},[20996,20998],{"id":5989,"depth":20997,"text":5989},3,{"id":10927,"depth":20997,"text":10927},{"id":14887,"depth":5953,"text":14888},{"id":19904,"depth":5953,"text":19905},{},"\u002Fblog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory",{"title":5980,"description":5952},"blog\u002F2026\u002F2026-06-06-proof-of-the-uat-by-weierstrass-theory","UAT（通用逼近定理）的核心要旨是：神经网络能逼近任何连续函数，而连续函数又可用多项式逼近（魏尔斯特拉斯定理）。",[5975,5974],"2026-06-06T00:00:00+08:00","-MwRVHIldXNGyyTz2c_FU7HHD2mDEdTWCl6xWAjBgLM",{"id":21010,"title":21011,"body":21012,"description":5952,"draft":5964,"enableComment":5964,"extension":5966,"image":5952,"important":5964,"location":5967,"meta":21325,"navigation":5965,"ogImage":5967,"path":21326,"seo":21327,"stem":21328,"summary":21329,"tags":21330,"time":21332,"__hash__":21333},"blog\u002Fblog\u002F2026\u002F2026-05-26-the-messianic-narrative-of-anthropic.md","The Messianic Narrative of Anthropic",{"type":8,"value":21013,"toc":21318},[21014,21018,21021,21024,21030,21033,21038,21041,21046,21053,21057,21065,21075,21080,21087,21090,21093,21096,21116,21119,21136,21139,21143,21148,21158,21161,21168,21171,21174,21178,21181,21191,21202,21207,21214,21217,21223,21240,21247,21250,21257,21263,21266,21270,21273,21287,21290,21293,21299,21302,21305,21308,21313],[30,21015,21017],{"id":21016},"judaism-and-the-messiah","Judaism and the Messiah",[14,21019,21020],{},"Within the theological framework of Judaism, the figure of the Messiah represents a systematic redemptive expectation gradually crystallized by the ancient Jewish people through prolonged suffering, exile, and doctrinal reflection. Its core impetus derives from a structural predicament: following the successive failures of monarchical authority and the Temple, the nation required a force transcending empirical reality to sustain identity and hope. After losing sovereign kingship, Temple sovereignty, and political freedom, a narrative capable of indefinitely deferred verification — yet perpetually furnishing meaning for action and ultimate hope — became an existential necessity. The Messiah constitutes the personified condensation of this narrative, bearing the weight of every unmet worldly aspiration: national independence, social justice, bodily resurrection, perpetual peace among nations, and the direct rule of the Divine.",[14,21022,21023],{},"Successive eras and communities have projected disparate contents into this \"formula\": the oppressed inserted armed liberation; the devout inserted Temple restoration; the afflicted inserted eschatological recompense; the guilt-ridden inserted vicarious atonement. Christianity elected the genetic strands of the \"Suffering Servant\" and the \"Heavenly Son of Man\" to define Jesus; mainstream Judaism, by contrast, ultimately rejected Jesus and preserved the yet-to-arrive, certain-to-come, thoroughly earthly Davidic Messiah.",[14,21025,21026],{},[21027,21028,21029],"strong",{},"Such is the genesis of the Messianic archetype within Judaism.",[14,21031,21032],{},"From the perspectives of comparative mythology and political psychology, this structure approximates what may be termed an archetypal narrative in human culture:",[11,21034,21035],{},[14,21036,21037],{},"The world descends into crisis;\nThe masses remain unaware;\nA minority attains true knowledge;\nFrom this minority a prophet emerges;\nThe prophet suffers doubt, incomprehension, and attack;\nThe prophet, having endured tribulation, ultimately acquires authority or influence;\nSalvation is delivered to the multitude, or catastrophe averted.",[14,21039,21040],{},"Viewed through this lens, the Messiah transcends the boundaries of a merely religious concept; it also constitutes one of the most ancient and influential archetypes in the human narrative repertoire:",[11,21042,21043],{},[14,21044,21045],{},"When the world descends into crisis, a chosen figure emerges, bearing a mission surpassing ordinary human capacity, ultimately delivering redemption and renewal.",[14,21047,21048],{},[21049,21050],"img",{"alt":21051,"src":21052},"Anthropic refused to sign an initiative recognizing open-weight AI models, deliberately maintaining distance and \"independence\" from other AI companies","https:\u002F\u002Fimage-assets.dreams.plus\u002F202607310100531.png",[30,21054,21056],{"id":21055},"the-biographical-background-of-the-amodei-siblings","The Biographical Background of the Amodei Siblings",[14,21058,21059,21060,21064],{},"The very name of the company is revealing in this regard: \"Anthropic,\" etymologically derived from the Greek ",[21061,21062,21063],"em",{},"anthrōpos"," — \"human being\" — signals their self-conception as a technology enterprise oriented toward human-centeredness and the destiny and welfare of all humankind. This, precisely, is the role they have long labored to inhabit.",[14,21066,21067,21068,21074],{},"It is by no means coincidental that the Amodei siblings' mother was a devout Jewish adherent. Their mother, ",[8501,21069,21073],{"href":21070,"rel":21071},"https:\u002F\u002Fgrokipedia.com\u002Fpage\u002FElena_Engel#ref-19",[21072],"nofollow","Elena Engel",", is a Jewish-American library project manager; their father, Riccardo Amodei, was an Italian leather artisan who suffered chronic illness and passed away when Dario was still young. The present author submits that there are reasonable grounds to suspect a correlation between Dario's preoccupation with the Messianic narrative and the circumstances of his familial upbringing. In Jewish-American households, domestic instruction frequently occupies a formative role.",[11,21076,21077],{},[14,21078,21079],{},"As a mother, Engel played a key role in fostering her children's intellectual curiosity and ethical outlook, instilling a strong sense of right and wrong as well as responsibility toward improving the world.",[14,21081,21082,21083,21086],{},"This educational paradigm, which internalizes the repair of the world (",[21061,21084,21085],{},"Tikkun Olam",", a core ethical concept in Judaism) as an individual vocation, is in its essence a secularized Messianic spirit: the conviction that human action, rather than pure divine intervention, can propel the world toward perfection. The premature death of the father and the absence of a stable masculine pillar in the developmental environment may have further intensified Dario's psychological need for a grand, determinate, and ultimate explanatory framework.",[14,21088,21089],{},"The Messianic narrative supplies precisely such a structure: it promises a definitive inflection point from chaos to order, from suffering to redemption. His technological optimism — and indeed his faith in the salvific potential of artificial general intelligence — may thus be understood not as a deduction from pure rationality, but as the projection, onto the technological age, of an ethical-redemptive complex molded by family.",[14,21091,21092],{},"Why, in the era of AGI, would a technology company so spontaneously adopt discursive structures that historically belonged to the domains of religion, philosophy of history, and eschatology?",[14,21094,21095],{},"The present author contends that traditional religion and eschatology principally address three questions:",[406,21097,21098,21104,21110],{},[409,21099,21100,21103],{},[21027,21101,21102],{},"Explaining everything",": Why is the world as it is? What are the root causes of suffering, chaos, and injustice?",[409,21105,21106,21109],{},[21027,21107,21108],{},"Promising salvation",": What constitutes the ultimate, perfect future state? (e.g., the Millennium, Heaven, the earthly Kingdom of God)",[409,21111,21112,21115],{},[21027,21113,21114],{},"Indicating the path",": Through what means (faith, spiritual practice, revolution) can that future be attained?",[14,21117,21118],{},"With the emergence of AGI, these three dimensions find precise correspondence:",[406,21120,21121,21126,21131],{},[409,21122,21123,21125],{},[21027,21124,21102],{},": The root of all human afflictions — disease, poverty, conflict, even mortality itself — lies in \"insufficient intelligence.\" Extant human cognition and conventional computation cannot process information of adequate complexity.",[409,21127,21128,21130],{},[21027,21129,21108],{},": AGI or ASI (artificial superintelligence) will constitute an omniscient, omnibenevolent, quasi-omnipotent \"godlike entity,\" resolving energy crises, climate change, and disease, and even enabling \"mind uploading\" and \"digital immortality.\"",[409,21132,21133,21135],{},[21027,21134,21114],{},": The technological trajectory — computational scaling, algorithmic breakthroughs, alignment research.",[14,21137,21138],{},"To be clear, we cannot, on this basis, assert that Anthropic's value system derives directly from familial background, nor can we ascertain whether it bears any relationship to Israel; nor can we simply reduce its AI safety philosophy to personal developmental history. Nevertheless, this formative milieu may provide a useful interpretive lens for understanding why Dario has long gravitated toward apocalyptic and safety-centric narratives.",[30,21140,21142],{"id":21141},"the-narrative-of-capital-apocalyptic-marketing","The Narrative of Capital: Apocalyptic Marketing",[11,21144,21145],{},[14,21146,21147],{},"AI is perilous; only we can save AI.",[14,21149,21150,21151,21154,21155,6349],{},"In the modern commercial system, the dynamic behavior of markets is not determined solely by fundamental variables but is, to a substantial degree, governed by two mutually interwoven psychological variables: ",[21061,21152,21153],{},"expectation"," and ",[21061,21156,21157],{},"sentiment",[14,21159,21160],{},"Expectation constitutes the rational anchor of capital allocation behavior — it is grounded in deductions concerning future cash flows, technological trajectories, or policy environments, forming a logical framework subject to periodic verification. Sentiment, by contrast, functions as the nonlinear accelerator of expectation realization: driven by collective psychological states such as greed and fear, it amplifies or distorts information, thereby engendering systematic deviations of price from intrinsic value.",[14,21162,21163,21164,21167],{},"It is worth noting that capital is, in its essence, ",[21061,21165,21166],{},"narrative-preferring",": it seeks out and rewards stories that simultaneously exhibit simplicity, grand vision, and verifiable milestones. Only such narratives can efficiently aggregate consensus within an environment of information asymmetry, converting dispersed individual beliefs into coherent asset-pricing behavior.",[14,21169,21170],{},"Put differently, commercial competition has, to a considerable extent, evolved into a contest of competence in managing expectations and steering sentiment. The former demands that enterprises continually honor their narrative commitments through clear milestones; the latter demands prudent calibration in response to excessive fluctuations in market psychology.",[14,21172,21173],{},"Whether a commercial narrative can sustain capital attraction, then, is determined not by its fantastical appeal, but by whether it can, across the dual dimensions of expectation and sentiment, both ignite the imagination and withstand the scrutiny of periodic factual verification.",[30,21175,21177],{"id":21176},"malefactors-and-prophets","Malefactors and Prophets",[14,21179,21180],{},"Against the backdrop of intensifying Sino-American antagonism, Anthropic has exploited a Cold War stereotype deeply embedded in modern Western discourse: the portrayal of China as an authoritarian, malevolent, despotic, and formidable power — a threat not merely to its own populace but to the entire democratic world. Anthropic, cast in the role of the minority \"prophet\" who has discerned the danger ahead of others, must take action; the \"democratic camp\" must retain AI supremacy and assume the mantle of guardianship. The company has thus fashioned itself as a democratic, progressive, safety-oriented AI enterprise.",[14,21182,21183,21184,6349],{},"In recent times, Anthropic has not spared its criticism of Chinese large language models, alleging that they have performed distillation upon its Claude series of models. This is documented in the February 2026 report ",[21061,21185,21186],{},[8501,21187,21190],{"href":21188,"rel":21189},"https:\u002F\u002Fwww.anthropic.com\u002Fnews\u002Fdetecting-and-preventing-distillation-attacks",[21072],"Detecting and Preventing Distillation Attacks",[14,21192,21193,21194,21201],{},"Implicit within this construction is a metaphor: a regime of the \"authoritarian and malevolent\" variety, such as China, allegedly lacks vigilance and rationality regarding AI, and will abuse or recklessly permit AI technologies to inflict harm. Following President Trump's visit to China in May 2026, Anthropic published a report titled ",[21061,21195,21196],{},[8501,21197,21200],{"href":21198,"rel":21199},"https:\u002F\u002Fwww.anthropic.com\u002Fresearch\u002F2028-ai-leadership",[21072],"2028: Two Scenarios for Global AI Leadership",", in which it stated:",[11,21203,21204],{},[14,21205,21206],{},"If the frontier is set by regimes that treat AI as an instrument of repression, military advantage over democracies, and domestic control, the transition is less likely to go well, for those regimes' own citizens or anyone else.\nHistorically, the reach of authoritarian rule has been limited by its dependence on human enforcers to carry out surveillance and repression. Powerful AI systems may remove that dependency, enabling automated repression on a far greater scale. For that reason, the prospect of the CCP leading in AI is among the greatest threats to a successful transition.",[14,21208,21209,21210,21213],{},"From this logic, the Chinese Communist Party is no longer merely a state regime but is recast as a ",[21061,21211,21212],{},"civilizational risk",". This, in fact, already exceeds the traditional narrative of interstate competition, and it is here that Anthropic's approach becomes most peculiar. For ordinary interstate competition is typically framed around American interests versus Chinese interests. Anthropic's narrative, however, operates on an elevated discursive plane: the interests of the free world versus the risk to humanity's shared future.",[14,21215,21216],{},"Herein lies an \"eschatological escalation.\" In the preceding century, the Cold War narrative held that a Soviet victory would threaten the \"free world.\" In the AI era, the narrative has mutated into: the triumph of the wrong power → the endangerment of the future of human civilization.",[14,21218,21219,21222],{},[21027,21220,21221],{},"AI has been invested with a status approaching that of a savior — or an apocalypse."," If AGI truly is what its proponents believe it to be — surpassing all humans in intelligence, commanding scientific research, commanding the economy, commanding military affairs — then whoever first possesses AGI no longer merely possesses a technology, but rather the capacity to define the civilization of the future. AI safety, national security, and civilizational security are thereby progressively collapsed into a single question.",[14,21224,21225,21226,21229,21230,21229,21233,21229,21236,21239],{},"AI is, moreover, naturally suited to bear this narrative, insofar as it simultaneously possesses four amplifiers: immense unknowns, world-transforming potential, apocalyptic risk scenarios, and soteriological technological vision. One consequently observes, throughout the Anthropic community and Dario's blog, an enthusiasm for discussing AI in language approaching the theological: ",[21061,21227,21228],{},"Superintelligence",", ",[21061,21231,21232],{},"Alignment",[21061,21234,21235],{},"the Control Problem",[21061,21237,21238],{},"Existential Risk",". While these concepts carry technical significance, they are, at the cultural level, readily mapped onto the traditional eschatological framework.",[14,21241,21242,21243,21246],{},"In religious allegory, the chain of ",[21061,21244,21245],{},"crisis → enemy → guardian → redemptive solution"," constitutes an ancient and highly efficient mode of propagation. From this vantage point, AI companies may not merely be discussing technical problems; they may also be participating in a larger political and religious narrative.",[14,21248,21249],{},"Viewed accordingly, the competition among AI companies is not only a contest of model capabilities but also a struggle for discursive authority: who is qualified to define the risks of the future, who is qualified to play the guardian, and who is qualified to furnish the path to redemption.",[14,21251,21252,21253,21256],{},"However, the present author submits that what warrants the greatest vigilance here is not conspiracy but ",[21061,21254,21255],{},"sacralization",". The critical point is not whether Anthropic deliberately exploits Cold War stereotypes. What merits closer attention is that when a single organization simultaneously holds the authority to define risk, to interpret technology, and to interpret morality, it readily becomes invested with a certain \"sacredness.\" And the catastrophic mass movements of history have, more often than not, been carried out precisely within the framework of such \"sacralizing narratives.\"",[14,21258,21259,21262],{},[21061,21260,21261],{},"We know what the risk is; we know what the future is; we know what the correct path is."," This structure is a frequent guest in the histories of religious organizations, revolutionary parties, and state apparatuses. But for it to appear within a major technology company, elevated to the status of core values, is comparatively rare.",[14,21264,21265],{},"Historical experience instructs us that any force claiming to have simultaneously grasped \"the future,\" \"the truth,\" and \"moral legitimacy\" ought to be subjected to sustained scrutiny and oversight.",[30,21267,21269],{"id":21268},"conclusion","Conclusion",[14,21271,21272],{},"If one could still debate whether Anthropic's earlier posturing toward China and the U.S. Department of Defense amounted to \"brand theater,\" the present author maintains that the releases of Mythos and Fable represent the eve of Anthropic's narrative bankruptcy.",[14,21274,21275,21276,21279,21280,21283,21284,6349],{},"From the foregoing analysis, it becomes apparent that the predominantly negative online reception of Anthropic may, in fact, serve its purposes rather well. A degree of negative appraisal is not necessarily a loss; it may instead reinforce the company's distinctive positioning. Anthropic is quite content to style itself as a \"prophet\" or \"Messiah\"-type figure. It is not merely selling a model; it is also selling an ",[21061,21277,21278],{},"identity"," — a unique symbolic marker within the capital market. Anthropic's conduct has, throughout, been reiterating a single refrain: ",[21061,21281,21282],{},"we are not like the other AI companies."," When a technology becomes sufficiently consequential, what companies compete on is no longer solely the product but rather the ",[21061,21285,21286],{},"civilizational role",[14,21288,21289],{},"More intriguingly, this touches upon an economic question: in the AI era, \"safety\" is itself being transformed into a form of scarce brand equity.",[14,21291,21292],{},"If everyone can train models, then \"who is fastest,\" \"who is cheapest,\" and \"who is safest\" all become dimensions of competition — and Anthropic manifestly seeks to occupy the third. In its imagined division of civilizational labor within the capital market, OpenAI is the innovator, Google is the technology titan, Meta is the open-source champion, and Anthropic is the guardian — all facing a common adversary: \"malevolent China.\" And once the discourse enters the register of \"who is qualified to define the risks of the future and who is qualified to represent the interests of humanity,\" the debate naturally acquires theological and political-theological inflections, ceasing to be a merely engineering matter.",[14,21294,21295,21296,6349],{},"Whether or not one endorses this image, it indisputably possesses strong market differentiation, and the narrative is undeniably compelling. It must, at this juncture, be conceded that Anthropic is indeed adept at ",[21061,21297,21298],{},"storytelling",[14,21300,21301],{},"Nevertheless, the present author retains reservations regarding the story Anthropic tells, and the sustainability of this narrative remains open to question. For under a liberal market regime, in no other domain, in no other historical period, in no other market, has there ever been a precedent of capital willingly accepting compromise with \"safety\" as the selling point. Capital is inherently profit-seeking; where the returns are sufficiently high, manufacturing and selling the rope with which one is to be hanged is by no means an uncommon occurrence. The so-called \"guardian narrative\" is perhaps more a matter of brand positioning than a principle for which one is genuinely prepared to bear costs. From the standpoint of political economy, what ultimately determines the strength of an organization's convictions is seldom what it writes in its blog posts, but rather what it sacrifices when interests and principles come into conflict.",[14,21303,21304],{},"The present author holds that, from a Chinese perspective, there is no need whatsoever to engage with Anthropic's theatrics. If one strips away Anthropic's narrative, examining it purely from the perspective of corporate behavior, it is, regardless of the narrative framework it deploys, fundamentally a commercial enterprise. Commerce is inseparable from customers, financing, and IPO listings. It cannot wholly detach itself from commercial interests.",[14,21306,21307],{},"From the perspective of American strategic circles, the core anxiety is not China's ethnic character but rather the prospect that a political system divergent from the liberal democratic tradition might acquire the capacity to define the technical rules of the future. If China were to lead in the AGI competition, the future world order might be shaped according to Chinese institutional preferences. What is truly feared here is not ethnic identity but institutional competition.",[14,21309,21310],{},[21027,21311,21312],{},"In brief: for the United States and the West, heirs to over a century of liberal tradition, the prospect of a political regime at variance with the liberal-democratic value system acquiring the capacity to define the technical rules of the future is entirely unacceptable. It is precisely this anxiety that furnishes Anthropic with the operational space to fish in troubled waters.",[14,21314,21315],{},[21061,21316,21317],{},"Fin.",{"title":5952,"searchDepth":5953,"depth":5953,"links":21319},[21320,21321,21322,21323,21324],{"id":21016,"depth":5953,"text":21017},{"id":21055,"depth":5953,"text":21056},{"id":21141,"depth":5953,"text":21142},{"id":21176,"depth":5953,"text":21177},{"id":21268,"depth":5953,"text":21269},{},"\u002Fblog\u002F2026\u002F2026-05-26-the-messianic-narrative-of-anthropic",{"title":21011,"description":5952},"blog\u002F2026\u002F2026-05-26-the-messianic-narrative-of-anthropic","The theatrics of Anthropic, led by Dario Amodei, exhibit the quintessential characteristics of the Judaic Messianic narrative, whose structure typically proceeds as follows: the world confronts an immense crisis of which the majority remains unaware; only a select few prophets perceive the danger; these prophets must be entrusted with authority; and they shall deliver salvation to the world.",[21331],"thoughts","2026-05-26T00:00:00+08:00","3-4yjjEMG-Kko0xLNzT5soCHNiPhWsylH9pagPo3Ji0",{"left":21335,"top":21335,"width":21336,"height":21336,"rotate":21335,"vFlip":5964,"hFlip":5964,"body":21337},0,24,"\u003Cg fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\">\u003Cpath d=\"m4.748 7.645l-.331 3.644c-.05.54-.074.811-.03 1.07a2 2 0 0 0 .238.655c.131.228.325.422.711.808l5.176 5.176c.787.787 1.18 1.18 1.636 1.328c.402.131.835.131 1.237 0c.456-.148.853-.544 1.645-1.336l3.96-3.96c.792-.792 1.187-1.188 1.336-1.644a2 2 0 0 0-.001-1.236c-.148-.457-.543-.853-1.335-1.645l-5.163-5.163c-.39-.39-.584-.584-.813-.716a2 2 0 0 0-.656-.238c-.26-.045-.535-.02-1.084.03l-3.63.33c-.944.086-1.417.129-1.787.335a2 2 0 0 0-.775.775c-.205.368-.248.838-.333 1.773z\"\u002F>\u003Cpath d=\"M9.713 9.713a1 1 0 1 0-1.415-1.414a1 1 0 0 0 1.415 1.414\"\u002F>\u003C\u002Fg>",{"left":21335,"top":21335,"width":21336,"height":21336,"rotate":21335,"vFlip":5964,"hFlip":5964,"body":21339},"\u003Cpath fill=\"none\" stroke=\"currentColor\" stroke-linecap=\"round\" stroke-linejoin=\"round\" stroke-width=\"2\" d=\"M12 7v5l3 3m6-3a9 9 0 1 1-18 0a9 9 0 0 1 18 0\"\u002F>",1789084287930]