Corollary 4

— Let E and F be two locally convex spaces, $\mathfrak{S}$ a cover of E consisting of bounded subsets. If E is barrelled and F Hausdorff and quasi-complete, the space $\mathscr{L}_{\mathfrak{S}}(E; F)$ is Hausdorff and quasi-complete.

In fact, every bounded and closed subset of $\mathscr{L}_{\mathfrak{S}}(E; F)$ is simply bounded (because $\mathfrak{S}$ is a cover of E), hence equicontinuous (III, p. 25, th. 1) and consequently is a complete subspace of $\mathscr{L}_{\mathfrak{S}}(E; F)$ because of prop. 11 (III, p. 22).