Proposition 12

— Let E be a bornological locally convex space, F a complete locally convex Hausdorff space and $\mathfrak{S}$ a family of bounded subsets of E containing the image of every sequence converging to 0. Then the space $\mathscr{L}_{\mathfrak{S}}(E; F)$ is complete.

Let $\Phi$ be a Cauchy filter in $\mathscr{L}_{\mathfrak{S}}(E; F)$. Then $\Phi$ is a Cauchy filter for the topology of simple convergence, hence converges in $F^E$; moreover, its limit $u$ is a linear mapping from E into F and $\Phi$ converges to $u$ uniformly on every set of $\mathfrak{S}$ (GT, X, § 1, No. 5, prop. 5). It follows that the image under $u$ of a sequence converging to zero is a sequence converging to zero, hence, that $u$ is continuous, since E is bornological (III, p. 11, prop. 1, (iii)).