Corollary 3

— Let E be a barrelled space, F a locally convex Hausdorff space and $\Phi$ a filter on $\mathscr{L}(E; F)$ which converges simply in E to a mapping u from E into F. If $\Phi$ contains a simply bounded subset of $\mathscr{L}(E; F)$, or if $\Phi$ has a countable base, then u is a continuous linear mapping from E into F and $\Phi$ converges uniformly to u on every precompact subset of E.

Suppose first that $\Phi$ contains a simply bounded set H ; since H is equicontinuous (th. 1), the corollary follows from the corollary of prop. 5 (III, p. 18). If $\Phi$ has a countable base, every elementary filter $\Psi$ associated with a sequence $u_n$ (GT, I, § 6, No. 8) which is finer than $\Phi$ is then simply convergent to u in E and it follows from cor. 2 that u is a continuous linear mapping from E into F, and that $\Psi$ converges to u for the topology of uniform convergence on precompact subsets of E. Consequently, the same holds for $\Phi$, since the latter is the intersection of elementary filters, each finer than $\Phi$ (GT, I, § 6, No. 8).

We observe that a filter on $\mathscr{L}(E; F)$ which converges simply and has a countable base does not necessarily contain a simply bounded set : to see this consider the example of the filter of neighbourhoods of 0 in $\mathscr{L}(K; F)$ when the topology of F is metrizable, but cannot be defined by a single norm.