Proposition 13

Let X be a locally compact topological space. The Banach space $\mathscr{C}_0(X)$ is an approximation space.

Suppose first that X is compact. Let $A\subset \mathscr{L}^f(\mathscr{C}(X))$ be the set of mappings of the form $f\mapsto u(f|X_0)$, where $X_0$ is a finite subset of X and $u:\mathscr{C}(X_0)\rightarrow \mathscr{C}(X)$ is a linear mapping of norm $\leqslant 1$. The set A is equicontinuous. By Lemma 5, the identity mapping of $\mathscr{C}(X)$ belongs to the closure of A for the topology of simple convergence on $\mathscr{C}(X)$. The proposition then results from Remark 3 of III, p. 14.

Pass to the general case. Let Y be the Alexandroff compactification of X and $\omega$ the point at infinity of Y (TG, I, p. 67). Identify $\mathscr{C}_0(X)$ with the set of elements of $\mathscr{C}(Y)$ vanishing at $\omega$. Then $\mathscr{C}(Y)$ is the topological direct sum of $\mathscr{C}_0(X)$ and the vector space of constant mappings on Y. Since $\mathscr{C}(Y)$ is an approximation space by what precedes, the space $\mathscr{C}_0(X)$ is an approximation space (Remark 1 of III, p. 14).