Exercise 9, § 3

Let $E$ and $F$ be two Hausdorff topological vector spaces, and $H$ an equicontinuous subset of $\mathscr{L}(E; F)$. Show that if there exists a countable total set in $E$, and if every bounded subset of $F$ is metrizable, then $H$ is metrizable for the topology of simple convergence in $E$. If moreover, every bounded subset of $F$ satisfies the first axiom of countability, then so does $H$.