Proposition 6

— *Let H be an equicontinuous subset of $\mathscr{L}(E; F)$. If F is metrizable and if there exists a countable total set in E, then the uniform structure on H of simple convergence in E is metrizable. If in addition, there exists a countable total set in F, then there exists a countable everywhere dense set in H (for the topology of uniform convergence on compact subsets of E).

Let $(a_n)$ be a total sequence in E. Then the mapping $u \mapsto (u(a_n))$ is an isomorphism from $\mathscr{L}(E; F)$, where $\mathscr{L}(E; F)$ has the uniform structure of simple convergence on the set of the $a_n$, onto a uniform subspace of $F^\mathbf{N}$. If F is metrizable (resp. metrizable and satisfies the first axiom of countability) then this is also true for $F^\mathbf{N}$ (GT, IX, § 2, No. 4, cor. 2 and § 2, No. 8, corollary), and the proposition follows from prop. 5 (III, p. 17).