Corollary 1

— Let E be a locally convex Hausdorff space, F a locally convex space, and $\mathfrak{S}$ a family of bounded subsets of E. If E is semi-complete, then every simply bounded subset of $\mathscr{L}(E; F)$ is bounded for the $\mathfrak{S}$-topology.

It is enough to apply th. 2, after replacing the sets of $\mathfrak{S}$ by their closed, convex, balanced envelopes, since this does not change the $\mathfrak{S}$-topology.

When E is semi-complete (for example quasi-complete), we can talk of the bounded subsets of $\mathscr{L}(E; F)$ without specifying the $\mathfrak{S}$-topology, since these are the same for all the $\mathfrak{S}$-topologies whenever $\mathfrak{S}$ is a cover of E.