Corollary 2

— (Banach-Steinhaus theorem). — Let E be a barrelled space, F a locally convex Hausdorff space, and let $(u_n)$ be a sequence of continuous linear mappings from E into F, which converges simply to a mapping u from E into F. Then $u \in \mathscr{L}(E; F)$, and $(u_n)$ converges to u uniformly on every precompact subset of E.

The sequence $(u_n)$ is, in fact, simply bounded, hence equicontinuous, and the corollary follows from the cor. of prop. 5 of III, p. 18.