Corollary 1

— *Let E be a locally convex metrizable space, and F a normed space. Suppose that E and F both satisfy the first axiom of countability. Then $\mathscr{L}(E; F)$ is the union of a countable family of equicontinuous subsets and there exists a countable set in $\mathscr{L}(E; F)$ which is dense for the topology of uniform convergence on precompact subsets of E.

Let B be the unit ball of F and $(V_n)$ a countable fundamental system of neighbourhoods of 0 in E. For every integer n, the set $H_n$ of all $u \in \mathscr{L}(E; F)$ such that $u(V_n) \subset B$ is equicontinuous and $\mathscr{L}(E; F)$ is the union of the $H_n$. The corollary then follows from prop. 6.*