Theorem 2

— Let E be a locally convex Hausdorff space, F a locally convex space and $\mathfrak{S}$ a family of closed, convex, balanced, bounded and semi-complete subsets of E (III, p. 7). Every simply bounded subset H of $\mathscr{L}(E; F)$ is bounded for the $\mathfrak{S}$-topology.

Let $A \in \mathfrak{S}$. The space $E_A$ is then a Banach space (III, p. 8, corollary), hence barrelled. On the other hand, the canonical image of H in $\mathscr{L}(E_A; F)$ is simply bounded, hence equicontinuous (III, p. 25, th. 1). Consequently, the set of all $u(x)$ for $u \in H$ and $x \in A$, is bounded in F, which proves that H is bounded for the $\mathfrak{S}$-topology.