习题 6, § 3

令$(E_i)_{i \in I}$为Hausdorff局部凸空间族，其中都不为0，令$E$为乘积空间$\prod_{i \in I} E_i$，$F$为*范数*空间。 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 $i \notin H$.)