Remark 1

— Let M be a complete uniform subspace of $\mathscr{L}_{\mathscr{S}}(E; F)$. For every set of bounded subsets $\mathscr{S}' \supset \mathscr{S}$ of E, the $\mathscr{S}'$-topology is finer than the $\mathscr{S}$-topology on $\mathscr{L}(E; F)$; on the other hand, there exists a fundamental system of neighbourhoods of 0 for the $\mathscr{S}'$-topology which are closed for the topology of simple convergence (III, p. 13, Remark 2), and a fortiori for the $\mathscr{S}$-topology. We conclude (GT, III, § 3, No. 5, cor. 1) that M is complete for the $\mathscr{S}'$-topology.