Exercise 5, § 1

Let E and F be two locally convex Hausdorff spaces, $\mathscr{S}$ a family of subsets of E. In order that the $\mathscr{S}$-topology on the space $\mathscr{L}(E; F)$ be compatible with the vector space structure, it is necessary (and sufficient, cf. III, p. 13, corollary) that every set of $\mathscr{S}$ is bounded in E.