Corollary

— Suppose F is Hausdorff. Let H be an equicontinuous subset of $\mathscr{L}(E; F)$. If a filter $\Phi$ on H converges simply to a mapping $u_0$ from E into F, then $u_0$ is a continuous linear mapping from E into F, and $\Phi$ converges uniformly to $u_0$ on every precompact subset of E.

The first assertion follows from prop. 4 (III, p. 16) and the second from prop. 5 (III, p. 17).