Corollary

Let $X$ be a paracompact, locally compact space; if $E$ is a quasi-complete locally convex space, then the space $\mathscr{K}(X; E)$ is quasi-complete.

It suffices, by virtue of Proposition 2, (ii), to note that for every compact subset $K$ of $X$, $\mathscr{K}(X, K; E)$ is a closed subspace of $C(K; E)$, which is quasi-complete since every bounded subset of $C(K; E)$ consists of functions taking values in a same bounded subset of $E$.