For $K$ compact metric, a family in $C(K)$ has compact closure in the sup norm exactly when it is uniformly bounded and equicontinuous.Suggest a slogan
Let \(K\) be a compact metric space, let \(C(K)\) be the space of continuous functions \(K\to\CC\) with the norm \(\norm{f}_\infty\coloneqq\sup_{x\in K}\abs{f(x)}\), and let \(\mcf\subseteq C(K)\). Then the closure of \(\mcf\) in \(C(K)\) is compact if and only if \(\mcf\) is uniformly bounded, that is, \(\sup_{f\in\mcf}\norm{f}_\infty<\infty\), and equicontinuous at every point of \(K\).
In particular, every sequence \((f_n)\) in \(C(K)\) that is uniformly bounded and equicontinuous at every point of \(K\) has a subsequence that converges uniformly on \(K\).
Since \(K\) is compact, a family \(\mcf\subseteq C(K)\) that is equicontinuous at every point of \(K\) is uniformly equicontinuous.
What is the Arzela-Ascoli theorem?
State the Arzela-Ascoli theorem.