Exercise 5, § 3

Let $(E_\alpha)_{\alpha \in A}$ be a family of locally convex spaces, $E$ a vector space (over the same field of scalars as the $E_\alpha$), and for each $\alpha \in A$, let $h_\alpha$ be a linear mapping from $E_\alpha$ into $E$. The space $E$ is assigned the finest locally convex topology for which the $h_\alpha$ are continuous (II, p. 27). For every $\alpha \in A$, let $\mathfrak{S}_\alpha$ be a family of bounded subsets of $E_\alpha$, and let $\mathfrak{S}$ be the union of the families $h_\alpha(\mathfrak{S}_\alpha)$ of bounded subsets of $E$. Under these conditions, show that, for every locally convex space $F$, the $\mathfrak{S}$-topology on $\mathscr{L}(E; F)$ is the coarsest topology for which the linear mappings $u \mapsto u \circ h_\alpha$ from $\mathscr{L}(E; F)$ into $\mathscr{L}_{\mathfrak{S}_\alpha}(E_\alpha; F)$ are continuous. In particular, if $E$ is the topological direct sum (II, p. 29, def. 2) of the family $(E_\alpha)_{\alpha \in A}$ (each $E_\alpha$ being identified with a subspace of $E$), then the product space $\prod_{\alpha \in A} \mathscr{L}_{\mathfrak{S}_\alpha}(E_\alpha; F)$ is canonically isomorphic to the space $\mathscr{L}_{\mathfrak{S}}(E; F)$, where $\mathfrak{S}$ is the union of the $\mathfrak{S}_\alpha$ in $\mathfrak{B}(E)$.