Exercise 10, § 7

a) Let E be a Hausdorff and complete locally convex space, $E'$ its dual, and ${E'}^* \supset E$ the algebraic dual of $E'$. Let A be a subset of E that is compact for the weakened topology $\sigma(E,E')$. Let $x$ be a point of ${E'}^*$ in the closure of the convex envelope C of A for the weak topology $\sigma({E'}^*,E')$. Show that if $(x_n')$ is a sequence of points of $E'$ that converges to $a' \in E'$ for the weak topology $\sigma(E',E)$, then $\lim_{n \to \infty} \langle x, x_n' \rangle = \langle x, a' \rangle$. (Note that $x$ is the barycenter of a measure of mass 1 on A, and apply Lebesgue’s theorem.)

b) Deduce from a) that if there exists in E a sequence of points that is dense for the original topology, then the restriction to every equicontinuous subset $H'$ of $E'$ of the mapping $x' \mapsto \langle x, x' \rangle$ is continuous for the weak topology $\sigma(E',E)$ (note that the topology induced on $H'$ by $\sigma(E',E)$ is metrizable). Deduce that in this case, necessarily $x \in E$ (TVS, IV, §5, No. 5); in other words, the closed convex envelope in E of a set that is compact for the weakened topology is again compact for that topology.

c) Extend the result of b) to the case that E is any quasi-complete Hausdorff locally convex space (Krein’s theorem). (First reduce to the case that E is complete by considering $\widehat{E}$; then note that by virtue of Eberlein’s theorem (TVS, IV, §5, No. 3, Th. 1), it suffices to prove that every sequence $(x_n)$ of points of C has a cluster point in E for $\sigma(E,E')$; this permits reducing to the case that there exists in E a sequence dense for the original topology.)