Exercise 16, § 4

Let $(E_i)_{i \in I}$ be an uncountable family of Hausdorff barrelled spaces, none of which are the point 0, and let $E = \prod_{i \in I} E_i$; then E is barrelled (IV, p. 14, corollary). Let G be the subspace of E consisting of all points $(x_i)$ such that $x_i = 0$ except for a countable number of indices. Every sequence of points of G which converges in E has a limit belonging to G, but G is dense in E. a) Show that every subset M of $G' = E'$, which is bounded for $\sigma(E', G)$ is contained in a finite product $\prod_{i \in H} E'_i$ (IV, p. 12, prop. 13), where H is a finite subset of I; consequently M is bounded for $\sigma(E', E)$. Deduce from this that G is barrelled. b) Let F be a subspace of E such that $G \subset F \subset E$ and such that G is a hyperplane (everywhere dense) in F ; F is barrelled (exerc. 15). Show that F is not bornological. (Argue by reductio ad absurdum ; if there were a convex, balanced and bounded set A in F such that G is an everywhere dense hyperplane in the normed space $F_A$ (III, p. 7), then there would exist a sequence of points of G converging to a point of F not belonging to G) (cf. IV, p. 52, exerc. 2). \*