Proposition 4

— Let X be a locally compact space, E a locally convex space; then, the space $\mathscr{K}(X;E)$ is dense in $C(X;E)$ for the topology of compact convergence.

For every compact set $K \subset X$, there exists a function $h \in \mathscr{K}(X;\mathbf{R})$ equal to 1 on K, by Lemma 1; for every function $f \in C(X;E)$ the function $hf$, which belongs to $\mathscr{K}(X;E)$, is equal to f on K, whence our assertion.