Exercise 4, § 2

Let $E$ be a Hausdorff and infra-barrelled locally convex space. a) Show that if the strong dual $E'_b$ of $E$ is bornological, then the completion $\hat{E}$ of $E$, identified with a vector subspace of ${E'}^*$ (III, p. 21, th. 2), is contained in the bidual $E''$ of $E$. b) We say that $E$ is distinguished if every subset of $E''$ which is bounded for the topology $\sigma(E'', E)$ is contained in the closure (for this topology) of a bounded subset of $E$. Show that for $E$ to be distinguished it is necessary and sufficient that its strong dual $E'_b$ is barrelled (cf. IV, p. 15, prop. 3).