From Hadamard to Langlands
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Talks about academic ancestors like Hadamard all the way to emeritus professor Langlands along the line of functional partially in Chinese for Ph.D students understanding part of ancient mathematics of China mixed with mainly Polish mathematician Banach theorems.
![有字经
使用教材
孙炯、王万义、郝建文：《泛函分析》，高等教育出版社，2010年第一版
参考资料
[1] 泛函分析（第二版），江泽坚等编，科学出版社。2004
[2] 夏道行, 吴卓人，严绍宗，舒五昌.实变函数论与泛函分析 下册.第二版北京:高等教育出版社.1985
[3] 夏道行，严绍宗等人. 泛函分析第二教程.北京:高等教育出版社.1983
我们俩的师傅：](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-7-320.jpg)
![Solving for the Extremal
? Differentiate I[?]:
? Set I[0] = 0 for the extremal, substituting terms for ? = 0 :
? Integrate second integral by parts:
? ? ? ?, ,
b b
x
x
xa a
wdI d F w FF x w w dx dx
w wd d
?
? ?? ?
? ?
? ?
? ?
? ?
?? ? ?? ? ?
? ? ? ?? ?
? ? ? ?w v x?
?
? ?
? ? ? ? ?x
x
w v x?
?
? ?
?
? ?0w v x
?
? ?
? ?
? ?
? ?
? ? ?0x
x
w v x
?
? ?
? ?
? ?
? ?
?
? ?0w u x? ?
? ?
? ?
?
? ?0x xw u x? ?
? ?
? ?
?
0
b
x
xa
dI F Fv v dx
u ud?
? ?
? ? ? ?? ?
? ?? ?
? ?
? ?? ?
? ?? 0
b b
x
xa a
F Fvdx v dx
u u
? ?? ?
? ?? ?
bb b b
x
x x x xa a aa
F F d F d Fv dx v vdx vdx
u u u udx dx
? ? ? ? ? ?
? ? ? ?? ?
? ? ? ?? ?
? ? ? ? ? ?
? ? ? ?? ? ? ?
? ? ? ?? ? ?
0
x
F
u
b b
a a
F dvdx vdx
u dx
? ??
? ?
?? ?
? ? ?
?? ? 0
x
F d F
u dx u
b
a
vdx
? ?? ?
? ?? ?? ?? ?? ?? ?
? ?
?
? ?
??](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-16-320.jpg)
![Cantor’s Set
0 11/3 2/3
Remove [1/3,2/3]
1/9 2/9 7/9 8/9
Remove [1/9,2/9] and [7/9,8/9]
0 11/3 2/3
Remove [1/3,2/3]](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-27-320.jpg)
![Cantor’s Set
0 11/3 2/3
Remove [1/3,2/3]
1/9 2/9 7/9 8/9
Remove [1/9,2/9] and [7/9,8/9]
Remove [1/27,2/27], [7/27,8/27], [19/27,20/27], and [25/27,26/27], and so on … What’s left?
雁过拔毛光身子了](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-28-320.jpg)
![We’ll call this set SVC (for Smith-Volterra-Cantor).
It has some surprising characteristics:
1. SVC contains no intervals - no matter how small a subinterval of [0,1] we take, there will be
points in that subinterval that are not in SVC. SVC is nowhere dense.
2. Given any collection of subintervals of [0,1], whose union contains SVC, the sum of the lengths of
these intervals is at least 1/2.
见面只分一半，留一条底裤](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-31-320.jpg)
![Volterra’s construction:
Start with the function F x? ? ?
x2
sin 1
x? ?, x ? 0,
0, x ? 0.
??
??
??
Restrict to the interval [0,1/8], except find the largest value of x on this interval at which F '(x) = 0, and keep F
constant from this value all the way to x = 1/8.](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-32-320.jpg)
![Volterra’s construction:
To the right of x = 1/8, take the mirror image of this function: for 1/8 < x < 1/4, and outside of
[0,1/4], define this function to be 0. Call this function .
f1 x? ?](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-33-320.jpg)
![Now we slide this function over so that the portion that is not identically 0 is in the interval
[3/8,5/8], that middle piece of length 1/4 taken out of the SVC set.](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-35-320.jpg)
![We follow the same procedure to create a new function, , that occupies the interval
[0,1/16] and is 0 outside this interval.
f2 x? ?
吃饱了撑的](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-36-320.jpg)
![Volterra’s function, V(x), is what we obtain in the limit as we do this for every interval removed from
the SVC set. It has the following properties:
1. V is differentiable at every value of x, and its derivative is bounded (below by –1.013 and above by
1.023).
2. If a is a left or right endpoint of one of the removed intervals, then the derivative of V at a exists
(and equals 0), but we can find points arbitrarily close to a where the derivative is +1, and points
arbitrarily close to a where the derivative is –1.
No matter how we partition [0,1], the pieces that contain endpoints of removed intervals
must have lengths that add up to at least 1/2.
The pieces on which the variation of V ' is at least 2 must have lengths that add up to at least
1/2.
左手不相信右手](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-38-320.jpg)
![Recall:
Vi ? sup
x?[ xi?1, xi ]
f x? ? ? inf
x?[xi?1,xi ]
f x? ?
Integral exists if and only if can be made as small as we wish by taking sufficiently small
intervals. Vi? xi ? xi?1? ?](https://image.slidesharecdn.com/fromhadamardtolanglands-190806062208/85/From-Hadamard-to-Langlands-39-320.jpg)
From Hadamard to Langlands
- 1. Functional Analysis 泛泛的函数分析 南方科技大学 史玉回实验室 跨年Mini Seminar 2018 ? 年幸月腊月到2019 ? 年正月杏月桃月 主讲人： Jun Steed ? Huang 资格证 ?
- 2. 研修（不是教学）大纲 TOC: Functional ? 《泛函分析》 讲座集中现代数学基础的重要基本∞抽象思想方法，以及工程难题的解答，内容可归纳为 ? 空间论 主要讲解空间的线性结构和度量结构以及两者的结合，其中包括度量空间、赋范线性空间和 内积空间，以及它们的拓扑结构。 ? 算子论 主要讲述对于线性泛函的“五大基本定理”：Riesz定理，开映像定理，闭图像定理，共鸣定 理，Hahn-Banach定理。 ? 谱论 主要讲解有界算子谱论及紧算子谱论的一些基本结论。 紧的含义：一尺之棰，日取其半，万世不竭。——《庄子·天下》，公元前300年 a平方b平方2ab 《周髀算经》345勾股定理，商高， 公元前1000年 代数 几何 C++OO 编 程 模 板 的 理 论 Eat tail when hungry：last to infinite!
- 7. 有字经 使用教材 孙炯、王万义、郝建文：《泛函分析》，高等教育出版社，2010年第一版 参考资料 [1] 泛函分析（第二版），江泽坚等编，科学出版社。2004 [2] 夏道行, 吴卓人，严绍宗，舒五昌.实变函数论与泛函分析 下册.第二版北京:高等教育出版社.1985 [3] 夏道行，严绍宗等人. 泛函分析第二教程.北京:高等教育出版社.1983 我们俩的师傅：
- 10. Steed的超方差与次相关 Matlab Code 的来龙去脉 https://ww2.mathworks.cn/matlabcentral/profile/authors/6205312-steed-huang
- 11. 从点线面体到超点超线超面超体
- 14. 变分学 ? Solutions to many physical problems require maximizing or minimizing some parameter: ? Distance ? Time ? Surface Area ? Parameter I dependent on selected path u and domain of interest D: ? Terminology: ? Functional – The parameter I to be maximized or minimized ? Extremal – The solution path u that maximizes or minimizes I ? ?, , x D I F x u u dx? ? 生活目标 评价函数 极大极小
- 15. Analogy to Calculus ? Single variable calculus: ? Functions take extreme values on bounded domain. ? Necessary condition for extremum at x0, if f is differentiable: ? ?0 0f x ?? ? Calculus of variations: ? True extremal of functional for unique solution u(x) ? Test function v(x), which vanishes at endpoints, used to find extremal: ? Necessary condition for extremal: ? ?, , b x a I F x w w dx?? ? ? ? ? ?? ? ? ? ? ?w x u x v x?? ? 0dI d? ?
- 16. Solving for the Extremal ? Differentiate I[?]: ? Set I[0] = 0 for the extremal, substituting terms for ? = 0 : ? Integrate second integral by parts: ? ? ? ?, , b b x x xa a wdI d F w FF x w w dx dx w wd d ? ? ?? ? ? ? ? ? ? ? ? ? ?? ? ?? ? ? ? ? ? ?? ? ? ? ? ?w v x? ? ? ? ? ? ? ? ?x x w v x? ? ? ? ? ? ?0w v x ? ? ? ? ? ? ? ? ? ? ? ?0x x w v x ? ? ? ? ? ? ? ? ? ? ? ?0w u x? ? ? ? ? ? ? ? ?0x xw u x? ? ? ? ? ? ? 0 b x xa dI F Fv v dx u ud? ? ? ? ? ? ?? ? ? ?? ? ? ? ? ?? ? ? ?? 0 b b x xa a F Fvdx v dx u u ? ?? ? ? ?? ? bb b b x x x x xa a aa F F d F d Fv dx v vdx vdx u u u udx dx ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ?? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ?? ? ? 0 x F u b b a a F dvdx vdx u dx ? ?? ? ? ?? ? ? ? ? ?? ? 0 x F d F u dx u b a vdx ? ?? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ??
- 17. The Euler-Lagrange Equation ? Since v(x) is an arbitrary function, the only way for the integral to be zero is for the other factor of the integrand to be zero. (Vanishing Theorem) ? This result is known as the Euler-Lagrange Equation ? E-L equation allows generalization of solution extremals to all variational problems. 0 x F d F u dx u b a vdx ? ?? ? ? ?? ?? ?? ?? ?? ? ? ? ? ? ? ?? x F d F u dx u ? ?? ? ? ? ? ? ?? ?
- 18. Functions of Two Variables ? Analogy to multivariable calculus: ? Functions still take extreme values on bounded domain. ? Necessary condition for extremum at x0, if f is differentiable: ? ? ? ?0 0 0 0, , 0x yf x y f x y? ? ? Calculus of variations method similar: ? ?, , , ,x y D I F x y u u u dxdy? ?? ? ? ? ? ? ?, , ,w x y u x y v x y?? ? ? ? ? ?, , , , yx x y x yD D wwdI d F w F F F x y w w w dxdy dxdy d d w w w ? ? ? ? ? ? ? ???? ? ? ? ? ? ? ?? ?? ?? ? ? ? ? ?? ? ?? ?? 0x y x yD D D F F F vdxdy v dxdy v dxdy u u u ? ? ? ? ? ? ? ? ??? ?? ?? 0 x yD F d F d F vdxdy u dx u dy u ? ?? ?? ?? ? ? ? ? ?? ?? ?? ? ? ?? ? ?? ?? ? ? ?? ? ?? x y F d F d F u dx u dy u ? ?? ?? ? ? ? ? ? ?? ? ? ? ?? ?? ? ? ?
- 19. Soap Film When finding the shape of a soap bubble that spans a wire ring, the shape must minimize surface area, which varies proportional to the potential energy. Z = f(x,y) where (x,y) lies over a plane region D The surface area/volume ratio is minimized in order to minimize potential energy from cohesive forces. ? ? ? ? ? ?? ? 2 2 , ; 1 x y D x y bdy D z h x A u u dxdy ? ? ? ? ??? 类似比表面积
- 20. Vito Volterra, 1881: There exists a function, F(x), whose derivative, F '(x), exists and is bounded for all x, but the derivative, F '(x), cannot be integrated. 特例1：震荡
- 21. The Fundamental Theorem of Calculus: 1. If then 2.If f does not have a jump discontinuity at x, then f x? ?a b ? dx ? F b? ?? F a? ?.F' x? ? ? f x? ? , d dx f t? ?dt ? f x? ?a x ? . If F is differentiable at x = a, can F '(x) be discontinuous at x = a?
- 22. F x? ? ? x2 sin 1 x? ?, x ? 0, 0, x ? 0. ?? ?? ?? If F is differentiable at x = a, can F '(x) be discontinuous at x = a? Yes!
- 23. F x? ? ? x2 sin 1 x? ?, x ? 0, 0, x ? 0. ?? ?? ?? F' x? ? ? 2xsin 1 x? ?? cos 1 x? ?, x ? 0.
- 24. F x? ? ? x2 sin 1 x? ?, x ? 0, 0, x ? 0. ?? ?? ?? F' x? ? ? 2xsin 1 x? ?? cos 1 x? ?, x ? 0. F' 0? ? ? lim h? 0 F h? ?? F 0? ? h ? lim h?0 h2 sin 1 h? ? h ? lim h?0 hsin 1 h? ?? 0 .
- 25. F x? ? ? x2 sin 1 x? ?, x ? 0, 0, x ? 0. ?? ?? ?? F' x? ? ? 2xsin 1 x? ?? cos 1 x? ?, x ? 0. F' 0? ? ? lim h? 0 F h? ?? F 0? ? h ? lim h?0 h2 sin 1 h? ? h ? lim h?0 hsin 1 h? ?? 0 . lim x?0 F' x? ? does not exist, but F' 0? ? does exist (and equals 0). 七上八下没个准
- 26. “Cantor’s Set” First described by H.J.S. Smith, 1875 Then by Vito Volterra, 1881 And finally by Georg Cantor, 1883 特例2：缺席了
- 27. Cantor’s Set 0 11/3 2/3 Remove [1/3,2/3] 1/9 2/9 7/9 8/9 Remove [1/9,2/9] and [7/9,8/9] 0 11/3 2/3 Remove [1/3,2/3]
- 28. Cantor’s Set 0 11/3 2/3 Remove [1/3,2/3] 1/9 2/9 7/9 8/9 Remove [1/9,2/9] and [7/9,8/9] Remove [1/27,2/27], [7/27,8/27], [19/27,20/27], and [25/27,26/27], and so on … What’s left? 雁过拔毛光身子了
- 29. Cantor’s Set 0 11/3 2/31/9 2/9 7/9 8/9 1? 1 3 ? 2 9 ? 4 27 ? 8 81 ??? ?? ?? ?? ? 1? 1 3 1? 2 3 ?? ?? ?? ?? ? 2 3 ?? ?? ?? ?? 2 ? 2 3 ?? ?? ?? ?? 3 ? ?? ?? ?? ?? ?? ?? ? 1? 1 3 ? 1 1? 2 3 ? 1? 1 ? 0 . What’s left has measure 0.
- 30. Create a new set like the Cantor set except the first middle piece only has length 1/4 each of the next two middle pieces only have length 1/16 the next four pieces each have length 1/64, etc. The amount left has size 1? 1 4 ? 2 16 ? 4 64 ? 23 44 ? ?? ?? ?? ?? ?? ?? ? 1? 1 4 1? 2 4 ?? ?? ?? ?? ? 2 4 ?? ?? ?? ?? 2 ? 2 4 ?? ?? ?? ?? 3 ? ?? ?? ?? ?? ?? ?? ? 1? 1 4 ? 1 1? 1 2 ? 1 2
- 31. We’ll call this set SVC (for Smith-Volterra-Cantor). It has some surprising characteristics: 1. SVC contains no intervals - no matter how small a subinterval of [0,1] we take, there will be points in that subinterval that are not in SVC. SVC is nowhere dense. 2. Given any collection of subintervals of [0,1], whose union contains SVC, the sum of the lengths of these intervals is at least 1/2. 见面只分一半，留一条底裤
- 32. Volterra’s construction: Start with the function F x? ? ? x2 sin 1 x? ?, x ? 0, 0, x ? 0. ?? ?? ?? Restrict to the interval [0,1/8], except find the largest value of x on this interval at which F '(x) = 0, and keep F constant from this value all the way to x = 1/8.
- 33. Volterra’s construction: To the right of x = 1/8, take the mirror image of this function: for 1/8 < x < 1/4, and outside of [0,1/4], define this function to be 0. Call this function . f1 x? ?
- 34. is a differentiable function for all values of x, but Volterra’s construction: f1 x? ? lim x?0? f1' x? ? and lim x? 1 4 ? f1' x? ? do not exist
- 35. Now we slide this function over so that the portion that is not identically 0 is in the interval [3/8,5/8], that middle piece of length 1/4 taken out of the SVC set.
- 36. We follow the same procedure to create a new function, , that occupies the interval [0,1/16] and is 0 outside this interval. f2 x? ? 吃饱了撑的
- 37. We slide one copy of into each interval of length 1/16 that was removed from the SVC set. f2 x? ?
- 38. Volterra’s function, V(x), is what we obtain in the limit as we do this for every interval removed from the SVC set. It has the following properties: 1. V is differentiable at every value of x, and its derivative is bounded (below by –1.013 and above by 1.023). 2. If a is a left or right endpoint of one of the removed intervals, then the derivative of V at a exists (and equals 0), but we can find points arbitrarily close to a where the derivative is +1, and points arbitrarily close to a where the derivative is –1. No matter how we partition [0,1], the pieces that contain endpoints of removed intervals must have lengths that add up to at least 1/2. The pieces on which the variation of V ' is at least 2 must have lengths that add up to at least 1/2. 左手不相信右手
- 39. Recall: Vi ? sup x?[ xi?1, xi ] f x? ? ? inf x?[xi?1,xi ] f x? ? Integral exists if and only if can be made as small as we wish by taking sufficiently small intervals. Vi? xi ? xi?1? ?
- 40. Conclusion: Volterra’s function V can be differentiated and has a bounded derivative, but its derivative, V ', cannot be integrated: d dx V x? ? ? v x? ?, but v t? ? dt ? V x? ?0 x ? ? V 0? ?. 需要从新定义积分（泛函）！
- 41. Lebesgue Measure
- 43. 研修（不是教学）大纲 TOC: Space ? 《泛函分析》 讲座集中现代数学基础的重要基本∞抽象思想方法，以及工程难题的解答，内容可归纳为 ? 空间论 主要讲解空间的线性结构和度量结构以及两者的结合，其中包括度量空间、赋范线性空间和 内积空间，以及它们的拓扑结构。 ? 算子论 主要讲述对于线性泛函的“五大基本定理”：Riesz定理，开映像定理，闭图像定理，共鸣定 理，Hahn-Banach定理。 ? 谱论 主要讲解有界算子谱论及紧算子谱论的一些基本结论。 紧的含义：一尺之棰，日取其半，万世不竭。——《庄子·天下》，公元前300年 a平方b平方2ab 《周髀算经》345勾股定理，商高， 公元前1000年 代数 几何 C++OO 编 程 模 板
- 44. 言归正传
- 47. Fixed Point Contraction Mapping
- 50. 研修（不是教学）大纲 TOC: Operator ? 《泛函分析》 讲座集中现代数学基础的重要基本∞抽象思想方法，以及工程难题的解答，内容可归纳为 ? 空间论 主要讲解空间的线性结构和度量结构以及两者的结合，其中包括度量空间、赋范线性空间和 内积空间，以及它们的拓扑结构。 ? 算子论 主要讲述对于线性泛函的“五大基本定理”：Riesz定理，开映像定理，闭图像定理，共鸣定 理，Hahn-Banach定理。 ? 谱论 主要讲解有界算子谱论及紧算子谱论的一些基本结论。 紧的含义：一尺之棰，日取其半，万世不竭。——《庄子·天下》，公元前300年 a平方b平方2ab 《周髀算经》345勾股定理，商高， 公元前1000年 代数 几何 C++OO 编 程 模 板
- 51. 搁颈别蝉锄表示定理
- 52. 热力学物理惭别补苍颈苍驳
- 55. 欧几里得: Euclid
- 56. 希尔伯特: Hilbert
- 59. Surjection
- 60. Category （纲） a nowhere dense set on a topological space is a set whose closure has empty interior.
- 61. Open Mapping Theorem Complete Measure Space is Dense Set
- 62. 有啥子用？
- 63. Direct Sum 图
- 67. Usage
- 70. Usage
- 71. 研修（不是教学）大纲 TOC: Spectrum ? 《泛函分析》 讲座集中现代数学基础的重要基本∞抽象思想方法，以及工程难题的解答，内容可归纳为 ? 空间论 主要讲解空间的线性结构和度量结构以及两者的结合，其中包括度量空间、赋范线性空间和 内积空间，以及它们的拓扑结构。 ? 算子论 主要讲述对于线性泛函的“五大基本定理”：Riesz定理，开映像定理，闭图像定理，共鸣定 理，Hahn-Banach定理。 ? 谱论 主要讲解有界算子谱论及紧算子谱论的一些基本结论。 紧的含义：一尺之棰，日取其半，万世不竭。——《庄子·天下》，公元前300年 a平方b平方2ab 《周髀算经》345勾股定理，商高， 公元前1000年 代数 几何 C++OO 编 程 模 板
- 72. Equations
- 75. 老老实实做人，规规矩矩办事
- 76. 厂辫别肠迟谤耻尘的严格定义
- 77. 先看清楚自己与厂别濒蹿的关系
- 78. 正则集合是开的: Open
- 80. 计算搁补诲颈耻蝉
- 81. 回到搁别补濒颈迟测世界
- 84. 会当临绝顶
- 87. Automorphic Form
- 89. L 函 子 朗兰兹函数（年青函数）的精神： L函数为数学对象的数值不变量 （通常为整数）的生成函数或无 限维李群的特征，可视为为复平 面上解析函数，这些数学对象可 以是表示 ，代数簇，流形，微分 算子等等，L（年青）函数期望 具有好的解析性质，比如函数方 程 乘积展开 零点和极点分布， 通过这些性质反映数学对象的算 术的，代数的，几何的，拓扑的， 组合的性质，以及它们的对称性 等等。
- 90. Pentagon 五角大楼
- 91. 自守钱贝流
- 92. 忘记嵌入日期的信
- 93. 再回到 Year 1880
- 94. 映入自相似笔别谤颈辞诲颈肠颈迟测
- 95. 抛开无穷尖点不算就是五指山
- 96. 搞来搞去，其实还是，喔靠，谱吗？
- 97. 最好压根没有钓鱼岛哎。。。
- 98. 革命纲领一统天下
- 100. 用斯猫加密吧，再见也不见也！