Exercise 12, § 3

Let E be a bornological locally convex space, and $\mathscr{S}$ be a family of bounded subsets of E containing the image of every sequence converging to 0. a) Show that for every locally convex space F, every bounded subset of $\mathscr{L}_\mathscr{S}(E; F)$ is equicontinuous. b) Show that if F is a Hausdorff and quasi-complete locally convex space, then the space $\mathscr{L}_\mathscr{S}(E; F)$ is quasi-complete.