Corollary

Let E and F be two locally convex spaces, H an equicontinuous subset of $\mathscr{L}(E; F)$. If F is Hausdorff and quasi-complete and if a filter $\Phi$ on H converges simply at all points of a total subset T of E, then there exists a continuous linear mapping u from E into F such that $\Phi$ converges uniformly to u on every precompact subset of E.

For, by virtue of prop. 5 (III, p. 17) $\Phi$ is a Cauchy filter for the uniform structure of precompact convergence in E; by prop. 11, the closure $\overline{H}$ of H in $\mathscr{L}_{pc}(E; F)$ is complete and so $\Phi$ converges uniformly on every precompact subset of E to a mapping $u \in \overline{H}$.