Corollary

— Let G be a Hausdorff locally convex space, and let $(u_n)$ be a sequence of linear mappings from F into G, converging simply to a mapping u from F into G. Then u is continuous, and the sequence $(u_n)$ converges to u uniformly on every precompact subset of F.

Since F is complete, the set of all $u_n$, which is bounded for the topology of simple convergence, is bounded in $\mathscr{L}_b(F; G)$ (III, p. 27, cor. 1). Since the space F is semi-barrelled (prop. 2), every countable and bounded subset of $\mathscr{L}_b(F; G)$ is equicontinuous by prop. 1 of IV, p. 21. Therefore the set of the $u_n$ is equicontinuous, and the corollary follows from III, p. 18, corollary.