Arzelà–Ascoli Theorem

Let $(K,d)$ be a compact metric space and consider $C(K,\mathbb{R})$ with the sup metric $d_\infty(f,g)=\sup_{x\in K}|f(x)-g(x)|.$

A subset $\mathcal{F}\subseteq C(K,\mathbb{R})$ is relatively compact if its closure in $(C(K,\mathbb{R}),d_\infty)$ is compact.

Arzelà–Ascoli Theorem (real-valued, compact metric domain): For $\mathcal{F}\subseteq C(K,\mathbb{R})$, the following are equivalent:

• $\mathcal{F}$ is relatively compact in $(C(K,\mathbb{R}),d_\infty)$.
• $\mathcal{F}$ is equicontinuous on $K$ and pointwise bounded on $K$.

Equivalently (sequential form): $\mathcal{F}$ is relatively compact if and only if every sequence in $\mathcal{F}$ has a uniformly convergent subsequence (with respect to $d_\infty$).

This theorem is the main compactness criterion for families of continuous functions and is central in existence proofs and approximation theory.