Exercise 4, § 4

Let $X$ be a locally compact space, $Y$ a paracompact locally compact space. Show that the canonical mapping of $\mathscr{K}(X \times Y; \mathbf{C})$ into $\mathscr{K}(X; \mathscr{K}(Y; \mathbf{C}))$ is an isomorphism of topological vector spaces.