Exercise 6, § 2

Let $F$ and $G$ be two vector spaces in separating duality. a) Show that the following properties are equivalent : $\alpha$ the space $F$ with a topology compatible with the duality between $F$ and $G$, is semi-reflexive; $\beta$ the space $G$, with $\tau(G, F)$, is barrelled. b) Show that the following properties are equivalent : $\alpha$ the space $F$, with $\tau(F, G)$, is reflexive;

β) the space G, with τ(G, F), is reflexive; γ) F and G are barrelled for τ(F, G) and τ(G, F) respectively. c) Show that every locally convex Hausdorff space E, with the topology τ(E, E') is isomorphic to the quotient of a semi-reflexive space F by a closed subspace. (By III, p. 44, exerc. 14, c), E' endowed with τ(E', E) is isomorphic to a closed vector subspace M of a Hausdorff barrelled space G; take F = G' endowed with τ(G', G) and use prop. 11 of IV, p. 10.) d) Let A be an infinite set with Card(A) > $\aleph_1$ (S, III, § 6, exerc. 10). In the product space P = $\mathbf{R}^A$, let E_0 denote the everywhere dense subspace consisting of all x = $(x_\alpha)_{\alpha \in A}$ such that $x_\alpha = 0$ except for a countable set of indices; the space E_0 is barrelled and so is the subspace E of P generated by E_0 and the point 1_A in P all whose coordinates are equal to 1 (III, p. 45, exerc. 16). Let B be a bounded subset in E_0, whose closure in P contains 1_A, and let J be a subset of A with cardinality $\aleph_1$; let pr_J denote the projection from P onto $\mathbf{R}^J$; show that there exists a subset B_J in B, with cardinality $\leq \aleph_1$, such that the closure of pr_J(B_J) in $\mathbf{R}^J$ contains 1_J; then the set J' ⊃ J of all indices $\alpha \in A$ such that for at least one point of B_J the coordinate with index $\alpha$ is $\neq 0$, has a cardinality equal to $\aleph_1$. We define J_0 = J and by induction $J_{n+1} = J'_n$, and put H = $\bigcup_n J_n$, whose cardinality is $\aleph_1$, and let $B_H = \bigcup B_{J_n}$; show that the closure of $B_H$ (and hence also that of B) contains the point $(1_H, 0) \in \prod^n \mathbf{R}^{H} \times \mathbf{R}^{A-H} = P$, which does not belong to E_0. Conclude that for every bounded and compact set C in E, C ∩ E_0 is again closed in E, and that E is not semi-reflexive. e) Deduce from d) that the dual E' of E (which can be identified with the dual P' = $\mathbf{R}^{(A)}$ of P) is not complete for the topology τ(E', E), inspite of being semi-reflexive (hence quasi-complete) for this topology (consider the linear form on E which is equal to 0 on E_0 and equal to 1 at the point 1_A, and use III, p. 21, th. 2).