Corollary

Let E and F be two locally convex spaces, $\mathfrak{S}$ a family of bounded subsets of E, and $\mathscr{L}(E; F)$ the vector space of continuous linear mappings from E into F. Then the $\mathfrak{S}$-topology* is compatible with the vector space structure of $\mathscr{L}(E; F)$ and is locally convex.*

It is enough to remark that if $u$ is a continuous linear mapping from E into F and M is a bounded subset of E, then $u(M)$ is bounded in F (III, p. 4, cor. 1).

Given two locally convex vector spaces E and F, and a family $\mathfrak{S}$ of bounded subsets of E, let $\mathscr{L}_\mathfrak{S}(E; F)$ denote the locally convex space obtained by assigning the $\mathfrak{S}$*-topology* to $\mathscr{L}(E; F)$.