Exercise 6, § 3

Let $(E_i)_{i \in I}$ be a family of Hausdorff locally convex spaces, none of which is 0, let $E$ be the product space $\prod_{i \in I} E_i$ and $F$ be a normed space. Show that there exists a canonical isomorphism from the space $\mathscr{L}(E; F)$ with the topology of bounded convergence (resp. of simple convergence, resp. of precompact convergence) onto the topological direct sum space of the spaces $\mathscr{L}(E_i; F)$, where each of these spaces is assigned the topology of bounded convergence (resp. of simple convergence, resp. of precompact convergence). (Observe that if $u$ is a continuous linear mapping from $E$ into $F$, then there exists a finite subset $H$ of $I$ such that $u^{-1}(0)$ contains the product of the $E_i$ for all indices $i \notin H$.)