Exercise 8, § 3

Let $E$ be a topological vector space, with a set of bounded subsets of $\mathfrak{S}$. Let $(G_i)_{i \in I}$ be a family of topological vector spaces, and $F$ be a vector space (over the same field of scalars as $E$ and the $G_i$); for every $i \in I$, let $g_i$ be a linear mapping from $F$ into $G_i$. Suppose $F$ is assigned the coarsest topology for which the $g_i$ are continuous. Show that the $\mathfrak{S}$-topology on $\mathscr{L}(E; F)$ is the coarsest topology for which the linear mappings $u \mapsto g_i \circ u$ from $\mathscr{L}(E; F)$ into $\mathscr{L}(E; G_i)$ are continuous. In particular, if $F = \prod_{i \in I} G_i$, the product space $\prod_{i \in I} \mathscr{L}(E; G_i)$ is canonically identified with $\mathscr{L}_{\mathfrak{S}}(E; F)$.