Exercise 2, § 3

Let $E$ and $F$ be two Hausdorff locally convex spaces, and $\mathfrak{S}$ a family of bounded subsets of $E$. a) Show that if $F$ is not just 0, then a necessary (and sufficient) condition for $\mathscr{L}_{\mathfrak{S}} (E ; F)$ to be Hausdorff is that the union of the sets of $\mathfrak{S}$ is total in $E$ (use Hahn-Banach th.). b) Suppose that $\mathfrak{S}$ is a cover for $E$. Show that there exists an isomorphism from $F$ onto a closed subspace of $\mathscr{L}_{\mathfrak{S}} (E ; F)$. Deduce that if $\mathscr{L}_{\mathfrak{S}} (E ; F)$ is quasi-complete, then $F$ is necessarily quasi-complete. c) Suppose that $\mathfrak{S}$ is a bornology adapted to $E$ (III, p. 3, def. 4). In order that $\mathscr{L}_{\mathfrak{S}} (E ; F)$ be metrizable, it is necessary and sufficient that $F$ is metrizable and that there exists a countable base (III, p. 1) for the bornology $\mathfrak{S}$. In order that the $\mathfrak{S}$-topology on $\mathscr{L}(E ; F)$ be defined by a single norm it is necessary and sufficient that the topology of $F$ can be defined by a single norm and that there exists a set $M \in \mathfrak{S}$ which absorbs every set of $\mathfrak{S}$.