Proposition 11

Let E and F be two locally convex spaces, $\mathscr{S}$ a cover of E consisting of bounded subsets. If F is Hausdorff and quasi-complete (III, p. 8), then every equicontinuous subset H of $\mathscr{L}(E; F)$ which is closed for the $\mathscr{S}$-topology is a complete uniform subspace of $\mathscr{L}_{\mathscr{S}}(E; F)$.

Since H is bounded in $\mathscr{L}_{\mathscr{S}}(E; F)$ (III, p. 22, prop. 9) and closed in $F^E$ for the $\mathscr{S}$-topology (III, p. 16, prop. 4), this follows from cor. 3 of GT, X, § 1, No. 5.