Proposition 5

— Let X be a locally compact space, E a real (resp. complex) locally convex space. For every compact subset K of X, the vector space $\mathscr{K}(X,K;\mathbf{R}) \otimes_{\mathbf{R}} E$ (resp. $\mathscr{K}(X,K;\mathbf{C}) \otimes_{\mathbf{C}} E$) (identified with a set of mappings of X into E, cf. A, II, §7, No. 7, Cor. of Prop. 15) is dense in $\mathscr{K}(X,K;E)$; the vector space $\mathscr{K}(X;\mathbf{R}) \otimes_{\mathbf{R}} E$ (resp. $\mathscr{K}(X;\mathbf{C}) \otimes_{\mathbf{C}} E$) is dense in $\mathscr{K}(X;E)$.

As the second assertion is an obvious consequence of the first, it suffices to prove the latter. We apply Lemma 2 with $\Phi$ reduced to a single element $f$ of $\mathscr{K}(X,K;E)$; then, for every $x \in X$,

where the $\varphi_j$ belong to $\mathscr{K}(X,K;\mathbf{R})$; since the mapping $x \mapsto \sum_{j=1}^n \varphi_j(x)f(x_j)$ may be canonically identified with the element $\sum_{j=1}^n \varphi_j \otimes f(x_j)$, this proves the proposition, by the definition of the topology of $\mathscr{K}(X,K;E)$.