Proposition 4

Let E, F be two locally convex spaces, and F be Hausdorff. Let the space $F^E$ of all mappings from E into F be assigned the topology of simple convergence. Then

(i) The set of linear mappings from E into F is closed in $F^E$.

(ii) If H is an equicontinuous subset of $\mathscr{L}(E; F)$, the closure $\overline{H}$ of H in $F^E$ is contained in $\mathscr{L}(E; F)$ and is equicontinuous.

We know that $\overline{H}$ is equicontinuous (GT, X, § 2, No. 3, prop. 6). It remains to prove the assertion (i). Let x, y be in E and $\lambda, \mu$ in K, and let $A(x, y, \lambda, \mu)$ be the set of all $u \in F^E$ such that

This set is closed in $F^E$ since the mapping $u \mapsto u(x)$ from $F^E$ into F is continuous for every $x \in E$ and since F is Hausdorff. But the set of linear mappings from E into F is equal to

Thus this set is closed in $F^E$.