Corollary 2

For every closed subset $S$ of $X$, every function $f \in \mathscr{K}(X, S; \mathbf{C})$ is the uniform limit of linear combinations $\sum_i \lambda_i \varphi_{K_i}$, where the $\lambda_i$ belong to $\mathbf{C}$ and the $K_i$ are compact subsets of $S$.

The set $\mathscr{A}$ of such linear combinations is a $\mathbf{C}$-algebra. Let $\Phi$ be the set of subsets $M$ of $X$ such that $\varphi_M \in \mathscr{A}$; $\Phi$ is thus a clan all of whose elements are subsets of $S$, containing the compact subsets of $S$, and $\mathscr{E}_C(\Phi) \subset \mathscr{A}$. It then suffices to apply Prop. 19 to the locally compact space $S$ and the clan $\Phi$.