Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 21H / a / Solution

For a compact space $K$, the Arzelà-Ascoli theorem says that a family $\mathcal{F}\subset C(K)$ is relatively compact in the supremum norm if and only if it is pointwise bounded and equicontinuous. In the usual formulation $K$ is compact Hausdorff; the elementary argument below also works for any compact $K$ for real-valued continuous functions.

Pointwise bounded means ${\sup }_{f\in \mathcal{F}}|f(x)|<\infty$ for every $x\in K$. Equicontinuity means that for every $x\in K$ and $\epsilon >0$ there is a neighborhood $U$ of $x$ such that $|f(y)-f(x)|<\epsilon$ for every $y\in U$ and $f\in \mathcal{F}$. Relative compactness means compactness of the closure in $C(K)$, where $\Vert f{\Vert }_{\infty }={\sup }_K|f|$. This is a complete metric vector space, since uniform limits of continuous functions are continuous. Equivalently, every sequence in the family has a uniformly convergent subsequence, with limit in $C(K)$, though not necessarily in the original family.