写在前面

本文是我在学习了点集拓扑学（我们跳过了专门讲述度量空间的部分）之后，学习泛函分析之前所写。而现在，尽管只是接触了极其有限的一点泛函，看来都是十分浅显的。所以本文便权当建立 Hugo 博客的练习文本好了。当然结果是非常失败的——我摆弄了几个小时的 \(\href{https://katex.org/}{KaTeX}\) Auto-render Extension，始终没能处理好转义字符的问题，最后还是用之前搞的 MathJax，把所有的 \(\)、\(\)、\, 全都加上一个斜杠才勉强实现现在的效果，但还是居然出现了输一个公式能渲染，再输一遍同样的公式就不能渲染了的情况（只能删掉），而这篇文章中甚至没出现过罗列公式和矩阵！考虑到后续还会有更多更复杂的数学公式要书写，我已经放弃用 markdown + Katex/MathJax，不如直接链接到 Latex 文档呢。（不会 html 和 css 导致的🙈）

另：vscode 的 Markdown Preview Enhanced 里的 Open in Browser 功能就很漂亮啊，不知道能不能拿来用。不过整个网站的宏观设置都用不了就是。

这里还是备忘一下现在效果的实现过程：

  1. themes/hugo-PaperMod-master/layouts/partials/extend_head.html 中添加：
    {{ if or .Params.math .Site.Params.math }}
{{ partial "math.html" . }}
{{ end }}
    这里以 “math” 为例，可以换成其他名字。
  2. 在上述文件夹下新建一个 math.html 文件，并添加：
    <!DOCTYPE html>  
<html lang="en">  
<head>  
    <meta charset="UTF-8">  
    <meta name="viewport" content="width=device-width, initial-scale=1.0">  
    <title>Math in Table</title>  
    <script type="text/javascript" async  
        src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js">  
    </script>
    <style>
        body {
           font-family: 'Times New Roman', Times, serif,'楷体' !important;
        }
    </style>
</head>  
<body>
    （这里还包含了字体信息）。
  3. 在需要用到数学公式的文档开头加上 math: true。

好了不说了，滚去学犯寒了（逃）。

Main Text

Introduction

The Arzelà–Ascoli theorem stands as a pivotal theorem in functional analysis. It provides a crucial link between the compactness of a function space and its properties of equicontinuity and uniform boundedness. This theorem, named after the Italian mathematicians Giuseppe Arzelà and Enrico Ascoli, has found applications in numerous areas of analysis, ranging from the existence of solutions to differential equations to complex analysis.

To understand the Arzelà–Ascoli theorem, it is essential to grasp the concept of a compact Hausdorff space. A Hausdorff space is a topological space where distinct points have disjoint neighborhoods, a fundamental property that ensures separation of points. Compactness, on the other hand, is a topological property that generalizes the notion of “closed and bounded” from Euclidean spaces to arbitrary topological spaces. Compact Hausdorff spaces, therefore, possess both the properties that ensures certain functions and sequences have convergent subsequences.

Theorem Statement and Proof

The original form of the theorem, discussed on closed interval on real numbers, is represented as follow:

Theorem 1. (Arzelà–Ascoli) Consider a sequence of real-valued continuous functions defined on a closed and bounded interval \([a,b]\) of the real line. If there exists a subsequence \(\{ f_{n}\}_{k\in \mathbb{N}}\) that converges uniformly, the sequence has such properties:

  1. Uniformly boundedness. \(\exists M>0,\, \forall n\in \mathbb{N},\, x\in [a,b],\, |f_n(x)|\leq M\);
  2. Uniformly equicontinuity. That is, \(\forall \varepsilon > 0,\, \exists \delta > 0,\, \forall n\in \mathbb{N},\, x,y\in [a,b],\, |x-y|<\delta \Rightarrow |f(x)-f(y)|<\varepsilon\).

The converse is also true.

Proof of this theorem is not very hard, so we omit it, as it can be cited from which of the general form we are going to talk about. Before that we should do some preparation for the topological space our theorem is based on.

Definition 1. (ring of continuous functions) Let \(X\) be a topological space and \(\mathscr{C}(X)\) be the function space consisting of all continuous functions from \(X\) into \(R\).

To see \(\mathscr{C}(X)\) as a topological space, we define a topology from a convergence point of view as follow:

Definition 2. (the topology induced by the uniform norm) In a normed linear space \(X\), convergence of a sequence \(\{ x_n\} \) to a point \(x\) is equivalent to \(\lim\limits_{n\to \infty}||x_n-x|| = 0\) (convergence in norm).


Now let’s restate the theorem in topologicla field:

Theorem 2. (Arzelà–Ascoli) Let \(X\) be a complete metric space, \(F\subseteq \mathscr{C}(X)\) (the continuous mapping space on \(X\)). Then \(F\) is sequentially compact if and only if:

  1. \(F\) is uniformly bounded, that is, \(\exists M>0,\, \forall f\in F,\, |f|\leq M\);
  2. \(F\) is uniformly equicontinuous, that is, \(\forall \varepsilon > 0,\, \exists \delta > 0,\, \forall f\in F,\, x,y\in X,\, d(x,y)<\delta \Rightarrow |f(x)-f(y)|<\varepsilon\).

To transform “compactness” into a kind of structure that is easier to deal with mathematically, “net” is introduced. It’s just like countably, infinitely segmenting the space. In math and real analysis, we have used this method inadvertently for a few times.

Lemma. (Hausdorff) \(X\) is a complete metric space. Then for \(M\subseteq X\), \(M\) is sequentially compact \(\Leftrightarrow\) \(\forall \delta >0\), there is a finite \(\delta\)-net of \(M\), i.e. \(\exists\) finite \(N\subseteq X,\, s.t.\, \forall x\in M,\, \exists y\in N,\, d(x,y)<\delta\).

“Net” is closely related to the diagonal rule. For sequence \(\{ x_i\} _{i=1}^{\infty}\subseteq A\), we have subsequence sequence based on finite net:

  • \(\{ x_i^{(1)}\} _{i=1}^{\infty} = \{ x_i\} _{i=1}^{\infty}\).
  • For finite \(\dfrac{1}{2}\)-net \(N_2\), \(\exists y_2\in N_2\), \(\{ x_i^{(2)}\} _{i=1}^{\infty} \triangleq \{ x\in \{ x_i^{(1)}\} _{i=1}^{\infty}\mid d(x,y_2)<\dfrac{1}{2}\} \) is infinite.

And so on, we got $$ \{ x_i^{(1)}\} _{i=1}^{\infty}\supseteq \{ x_i^{(2)}\} _{i=1}^{\infty}\supseteq \cdots $$

Aiming to choose one element of each subsequence to form a new subsequence, with order ensured, we extract it on the diagonal line: \(\{ x_i^{(i)}\} _{i=1}^{\infty}\). Because $$ \forall k>0,\, \forall m,n>k,\, d(x_k^{(m)},x_k^{(n)})\leq d(x_k^{(m)},y_k) + d(x_k^{(n)},y_k)<\dfrac{2}{k} $$

\(d(x_k^{(m)},x_k^{(n)})<\dfrac{2}{k}\). Therefore \(\{ x_i^{(i)}\} _{i=1}^{\infty}\) is Cauchy sequence, which is a convergent subsequence of \(\{ x_i\} _{i=1}^{\infty}\).

In essence, net is a promotion of limit. For topological space \(X\), a net is a mapping from a directed set \(D\) into \(X\). A net \((x_i)_{i\in D}\) is called converging to \(x\in X\) when and only when \(\forall\) neibourhood \(U\) of \(x\), \(\exists I\in D,\, \forall i\geq I, x_i\in U\). A directed set is, a preordered set \((D,\leq)\) which satisfies that \(\forall i,j\in D,\, \exists k\in D,\, s.t.\, i\leq k,j\leq k\).

Proof. Necessity. Through a finite \(\delta\)-net, uniformly boundedness is obvious.

For uniformly equicontinuity, \(\forall \varepsilon > 0\), make use of \(\dfrac{\varepsilon}{3}\)-net \(N_{\varepsilon}\). \(\forall g\in N_{\varepsilon}\), \(|f(x)-g(x)|\leq \varepsilon\). Observing that $$ |f(x)-f(y)|\leq |f(x)-g(x)|+|f(y)-g(y)|+|g(x)-g(y)|<\dfrac{2}{3}\varepsilon + |g(x)-g(y)| $$

As there are only finite \(g\)s, \(\exists \delta > 0,\, s.t.\) \(|g(x)-g(y)|\) is sufficiently small. Therefore uniformly equicontinuity is proved.

Sufficiency. Conversely, \(\forall \varepsilon > 0\), we could find a finite \(\delta\)-net \(N_\varepsilon\) that $$ \forall f\in F,\, x,y\in X,\, d(x,y)<\delta \Rightarrow |f(x)-f(y)|<\dfrac{\varepsilon}{3} $$

Consider the mapping $$ T:F\longrightarrow \mathbb{R}^{|N_\varepsilon|}\qquad f\longmapsto (f(n))_{n\in N} $$

Obviously, \(\textup{Im}T\) is bounded. Account for the sequencial compactness of \(\mathbb{R}^{|N_\varepsilon|}\)’s bounded subset, we have \(\dfrac{\varepsilon}{3}\)-net, denoted as \(\{ Tf_1,\cdots,Tf_m\} \). Then \(\forall f\in F,\, |f(x)-g(x)|<\varepsilon\). Q.E.D.

As we can see, it is a common technique to transform the distance between two points in the space to be considered into a more easily examined distance between two points in the space to be considered, and the distance between the two points in the space to be considered, by a reduction method similar to the triangle inequality (actually a “quadrilateral”).

Promotion

The condition of the theorm could be weakened. Here we list two directions of the promotion:

  1. Onto non-compact metric space.
  2. Onto discontinuous mapping.

Application Example: ODE

Any thoughtful student of mathematics should have an idea of the extensibility of the existence and uniqueness of solutions to ordinary differential equations when first exposed to it, which seems so ingenious that it seems like a small application of a broader conclusion.

Actually, Lipchitz condition $$ \exists L, \forall n, \forall x,y\in [a,b], |f_n(x)-f_n(y)|\leq L|x-y| $$

is a direct result of uniform boundedness of the derivatives, letting \(L=\sup{f’_n(x)}\). So in addition, given \(\varepsilon > 0\), let \(\delta = \dfrac{\varepsilon}{2L}\) to verify equicontinuity, the conclusion of the theorem is apparent.

Conclusion

From the discussion above, we have got a deeper understanding of the concept “convergence” in foundational analysis, and built a bridge between compact Hausdorff spaces, the very basic topological concept, and the analytical properties of function spaces. The importance of the Arzelà–Ascoli theorem lies in its wide-ranging applications. It is a crucial ingredient in the proof of the Peano existence theorem for solutions to ordinary differential equations. It also plays a vital role in the proof of Montel’s theorem in complex analysis. Furthermore, the theorem has found applications in areas such as approximation theory, functional equations, and the study of integral equations. Its depth and breadth of applications have made it a cornerstone of modern analysis.

Reference

[1] Ascoli, G., “Le curve limite di una varietà data di curve”, Atti della R. Accad. Dei Lincei Memorie della Cl. Sci. Fis. Mat. Nat., 18 (3): 521–586
[2] Dunford, Nelson; Schwartz, Jacob T. (1958), Linear operators, volume 1, Wiley-Interscience
[3] James R. Munkres. Topology: a first course[M]. Englewood Cliffs: Prentice-Hal, Inc., 1975