Proposition 2

— Let X be a locally compact space, E a Hausdorff locally convex space.

(i) If E is a Fréchet space, then the space $\mathscr{K}(X; E)$ is barreled.

(ii) If X is paracompact then, for every bounded set B in $\mathscr{K}(X; E)$, there exists a compact subset K of X such that $B \subset \mathscr{K}(X, K; E)$.

Suppose E is a Fréchet space. Then, for every compact subset K of X, $\mathscr{K}(X, K; E)$ is a Fréchet space, hence is barreled, and one knows that a direct limit of barreled spaces is barreled (TVS, III, §4, No. 1, Cor. 3 of Prop. 3), whence (i).

If X is paracompact, one knows (GT, I, §9, No. 10, Th. 5) that X is the sum of a family $(X_\lambda)_{\lambda \in L}$ of locally compact spaces that are countable at infinity; thus (Prop. 1, (iii)), $\mathscr{K}(X; E)$ is the topological direct sum of the family of subspaces $\mathscr{K}(X_\lambda; E)$ ($\lambda \in L$). By virtue of the characterization of the bounded sets in a topological direct sum (TVS, III, §1, No. 4, Prop. 5), every bounded set in $\mathscr{K}(X; E)$ is contained in the sum of a finite number of subspaces $\mathscr{K}(X_\lambda; E)$, and it will suffice to prove that every bounded set in $\mathscr{K}(X_\lambda; E)$ is contained in a subspace $\mathscr{K}(X_\lambda, K_\lambda; E)$, with $K_\lambda$ compact in $X_\lambda$. We are thus reduced to the case that X is countable at infinity, in other words is the union of a sequence of relatively compact open sets $U_n$ such that $\overline{U}_n \subset U_{n+1}$ (GT, I, §9, No. 9, Prop. 15). But then $\mathscr{K}(X; E)$ is the strict direct limit of the sequence of spaces $\mathscr{K}(X, \overline{U}_n; E)$, whence the assertion (ii) (TVS, III, §1, No. 4, Prop. 6).

We shall say that a subset H of $\mathscr{K}(X; E)$ is strictly compact if it is compact and if there exists a compact subset K of X such that $H \subset \mathscr{K}(X, K; E)$. It follows at once from Proposition 2 that if X is a paracompact locally compact space and if E is Hausdorff, then every compact set in $\mathscr{K}(X; E)$ is strictly compact. One can give examples of locally compact

3 Dénombrable à l'infini, also translated as "σ-compact" (GT, I, §9, No. 9, Def. 5).

spaces $X$ (not paracompact) such that there exist sets in $\mathscr{K}(X; \mathbf{R})$ that are compact but not strictly compact (Exercises 3 and 4).

We recall that, by virtue of Ascoli’s theorem (GT, X, §2, No. 5, Cor. 3 of Th. 2), a strictly compact subset $H$ of $\mathscr{K}(X; E)$ contained in $\mathscr{K}(X, K; E)$ is characterized by the following conditions: $1^\circ$ it is closed; $2^\circ$ it is equicontinuous; $3^\circ$ for every $x \in K$, the set $H(x)$ is relatively compact in $E$.