Proposition 1

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

(i) The locally convex space $\mathscr{K}(X; E)$ is Hausdorff. For every compact subset $K$ of $X$, the topology on $\mathscr{K}(X, K; E)$ induced by that of $\mathscr{K}(X; E)$ is the topology of uniform convergence in $K$, and each of the subspaces $\mathscr{K}(X, K; E)$ is closed in $\mathscr{K}(X; E)$.

(ii) If $E$ is the product of a finite number of locally convex spaces $E_i$ ($1 \leq i \leq n$), then the mapping $f \mapsto (\mathrm{pr}_i \circ f)$ is an isomorphism of the space $\mathscr{K}(X; E)$ onto the product space $\prod_{1 \leq i \leq n} \mathscr{K}(X; E_i)$.

(iii) If $X$ is the sum of a family $(X_\lambda)_{\lambda \in L}$ of locally compact spaces, then the mapping $f \mapsto (f|X_\lambda)_{\lambda \in L}$ is an isomorphism of the space $\mathscr{K}(X; E)$ onto the topological direct sum space of the family $(\mathscr{K}(X_\lambda; E))_{\lambda \in L}$.

(i) Note that, on $\mathscr{K}(X; E)$, the topology of uniform convergence in $X$ is compatible with the vector space structure of $\mathscr{K}(X; E)$ because, for every $f \in \mathscr{K}(X; E)$, with (compact) support $S$, the set $f(X) = f(S) \cup \{0\}$ is compact, hence bounded in $E$ (TVS, III, §3, No. 1, Prop. 1). Since this topology $\mathscr{T}_0$ is locally convex and induces on each $\mathscr{K}(X, K; E)$ the topology of uniform convergence in $K$, the same is true of the direct limit topology $\mathscr{T}$ on $\mathscr{K}(X; E)$ (TVS, II, §4, No. 4, Remark); moreover, $\mathscr{T}$ is finer than $\mathscr{T}_0$ and $\mathscr{T}_0$ is Hausdorff, therefore $\mathscr{T}$ is Hausdorff. Finally, suppose that a function $f \in \mathscr{K}(X; E)$ belongs to the closure of $\mathscr{K}(X, K; E)$; by the definitions, there exists a compact subset $K' \supset K$ of $X$ such that $f \in \mathscr{K}(X, K'; E)$. By the foregoing, $f$ belongs to the closure of $\mathscr{K}(X, K; E)$ in the space $\mathscr{K}(X, K'; E)$, hence belongs to $\mathscr{K}(X, K; E)$.

(ii) The criterion for continuity in a direct limit (TVS, II, §4, No. 4, Prop. 5) shows at once that the mapping $f \mapsto (\mathrm{pr}_i \circ f)$ is continuous and that the same is true of the inverse mapping (for the latter, it suffices to note that if, for every function $f_i \in \mathscr{K}(X; E_i)$, one denotes by $f'_i$ the mapping

² Limite inductive, translated as "direct limit" in S, A and GT.

of X into E such that $\mathrm{pr}_i \circ f'_i = f_i$ and $\mathrm{pr}_j \circ f'_i = 0$ for $j \neq i$, then each of the mappings $f_i \mapsto f'_i$ is continuous).

(iii) Each compact subset K of X intersects only the $X_\lambda$ of a finite sub-family $(X_\lambda)_{\lambda \in H}$ of $(X_\lambda)_{\lambda \in L}$, and it is immediate that if one sets $K_\lambda = K \cap X_\lambda$ for $\lambda \in H$, the mapping $f \mapsto (f|X_\lambda)_{\lambda \in H}$ is an isomorphism of $\mathscr{K}(X, K; E)$ onto $\prod_{\lambda \in H} \mathscr{K}(X_\lambda, K_\lambda; E)$. Conversely, for every function $f_\lambda \in \mathscr{K}(X_\lambda; E)$, let $f''_\lambda$ be the mapping of X into E such that $f''_\lambda|X_\lambda = f_\lambda$ and $f''_\lambda|X_\mu = 0$ for $\mu \neq \lambda$; it is immediate that the mapping $f_\lambda \mapsto f''_\lambda$ of $\mathscr{K}(X_\lambda; E)$ into $\mathscr{K}(X; E)$ is continuous. The assertion (iii) follows from these remarks and the criterion for continuity in direct limits (TVS, II, §4, No. 4, Prop. 5).