Theorem 1

— Let E be a barrelled space, F a locally convex space. Every simply bounded subset H of $\mathscr{L}(E; F)$ is equicontinuous.

For, let $p$ be a continuous semi-norm on F; let $q = \sup_{u \in H} (p \circ u)$. Since H is simply bounded, we have $q(x) < +\infty$ for all $x \in E$ and $q$ is a lower semi-continuous semi-norm, being the finite superior envelope of continuous semi-norms. Since E is barrelled, $q$ is a continuous semi-norm and therefore H is equicontinuous.