Exercise 14, § 4

a) Let E be a Hausdorff locally convex space, L a subspace of E with finite codimension, and T a barrel in L. Show that there exists a barrel $T'$ in E such that $T' \cap L = T$ (show that we can take for $T'$ the sum of the closure $\overline{T}$ of T in E and of a finite dimensional compact convex set). b) Let E be a barrelled space, L a subspace of E, which has a complement having a countable basis. Show that L is barrelled (use a) and exerc. 13, b)). \* c) Let E be a Hausdorff locally convex space; its completion $\hat{E}$ can be identified with a closed

EXERCISES

subspace of a barrelled space F, which is the product of a family of Fréchet spaces (II, p. 5, prop. 3 and IV, p. 14, corollary). Let $(e_\alpha)_{\alpha \in A}$ be the basis of a complement in F of the subspace E, and let $H_\alpha$ be the hyperplane in F generated by E and the $e_\beta$ for indices $\beta \neq \alpha$; by b), $H_\alpha$ is a barrelled space. For every $x \in E$, let $u(x)$ be the point of the barrelled space $G = \prod_{\alpha \in A} H_\alpha$ (sub-space of $F^A$) all whose coordinates are equal to $x$; $u$ is an isomorphism from E onto the subspace $\Delta \cap G$, where $\Delta$ is the diagonal in $F^A$. Show that $u(E) = \Delta \cap G$ is closed in G, and consequently that every Hausdorff locally convex space is isomorphic to a closed subspace of a Hausdorff barrelled space. \*