Exercise 16, § 1

Let F and G be two vector spaces in separating duality, the set of all linear forms $x'$ on F which are bounded an every subset of F bounded for $\sigma(F, G)$ is called the enclosure of G in the algebraic dual F* of F; this is a vector subspace $\tilde{G}$ of F*, which is the dual of F when F is assigned the topology of the bornological space associated (III, p. 40, exerc. 1) with one of the topologies compatible with the duality between F and G. Then G is said to be enclosed in F* if $\tilde{G} = G$. a) Let M be a closed vector subspace of F for the topology $\sigma(F, G)$. Show that if G is enclosed in F*, and if F/M is assigned the topology $\sigma(F/M, M^\circ)$, then its dual is enclosed in (F/M)*. b) Let E be a Hausdorff locally convex space, E' its dual. For E to be bornological, it is necessary and sufficient that its topology is identical with $\tau(E, E')$ and that E' is enclosed in E*. c) Let $(E_i)_{i \in I}$ be a family of Hausdorff locally convex spaces, and F the direct sum space of the $E_i$, endowed with the topology defined in I, p. 24, exerc. 14. Show that the dual of F is canonically isomorphic to the subspace of the product $\prod_{i \in I} E'_i$ of the duals of the $E_i$, consisting of all families $(x'_i)$ such that $x'_i = 0$ except for a countable number of indices. (Let $V_i$ be an arbitrary neighbourhood of 0 in $E_i$. Show that if $x'_i \neq 0$ for an uncountable set of indices, then there exists a number $\alpha > 0$ and an uncountable set $H \subset I$ such that there exists $x_i \in V_i$ for which $\langle x_i, x'_i \rangle \geq \alpha$ for all $i \in H$; conclude that $(x'_i)$ cannot belong to F'.) d) Show that if I is uncountable then F' is not enclosed in F*; if we take $E_i = \mathbf{R}$ for all $i \in I$, then F' endowed with the strong topology, is not complete, and there exist strongly bounded subsets in F' which are not weakly relatively compact.