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Section 7.5 Equicontinuity
Let \(X\) be a compact metric space and let \(C(x)\) be viewed as a metric space with the metric \(d_{\infty}\text{.}\)
Fact 7.21 .
Closed bounded sets in \(C[0,1]\) need not be compact.
Proposition 7.22 .
For a compact metric space \(X\text{,}\) TFAE:
There is a replacement for the Heine-Borel Thm for \(C(X)\) known as Arzelá-Ascoli Thm. Definition first
Definition 7.23 .
A set \(\mathcal{F}\subseteq C(x)\) is called (uniformly) equicontinuous if for all \(\varepsilon>0\) there exists a \(\delta>0\) such that for all \(x,y\in X\) and for all \(f\in\mathcal{F}\text{,}\) \(d(x,y)<\delta\implies\left|{f(x)-f(y)}\right|<\varepsilon\) (so \(\delta\) doesn't depend on \(f!\) ).
We could define \(\mathcal{F}\) to be equicontinuous at \(x\in X\) if for all \(\varepsilon>0\) exists \(\delta_x>0\) such that for all \(y\in X\) and all \(f\in\mathcal{F}\text{,}\) \(d(x,y)<\delta_x\implies \left|{f(x)-f(y)}\right|<\varepsilon\text{,}\) and pointwise equicontinuous if \(\mathcal{F}\) is equicontinuous at each point in \(X\) (so \(\delta\) depends on \(x\) but not \(f\) ).
Example 7.25 .
If \(\mathcal{F}\) is finite, then \(\mathcal{F}\) is equicontinuous iff each \(f\in\mathcal{F}\) is uniformly continuous (take \(\delta=\min_{f\in\mathcal{F}}\delta_f\) in the notation above).
Exploration 7.26 .
Suppose \(f_n,f:X\rightarrow\mathbb{C}\) are uniformly continuous and \(f_n\rightarrow f\) uniformly. Then \(\mathcal{F}=\{{f_n}\,:\,{n\geq 1}\}\cup\{{f}\}\) is equicontinuous.
Theorem 7.27 . Arzelá-Ascoli Theorem.
Let \(X\) be a compact metric space and \(\mathcal{F}\subseteq C(X)\text{.}\) Then
\(\mathcal{F}\) is compact iff \(\mathcal{F}\) is closed, bounded, and equicontinuous.
\(\mathcal{F}\) is totally bounded iff \(\mathcal{F}\) is bounded and equicontinuous.
