Exercise 11, § 2

Let E be a Hausdorff locally convex space such that the strong dual E'_b of E is semi-reflexive. a) Show that, on every strongly bounded subset of E', the topologies induced by $\sigma(E', E'')$ and $\sigma(E', E)$ are identical. b) Deduce from a) that E is infra-barrelled for the topology τ(E, E') (cf. IV, p. 52, exerc. 1) and that, if $\hat{E}$ is its completion for this topology, and is identified with a subset of E'* (III, p. 21, th. 2), then $E'' \subset \hat{E}$. In particular, if E is quasi-complete for τ(E, E'), then E is reflexive for this topology.

¹ On the other hand, a closed subspace of a Montel space need not be infrabarrelled, and the quotient of a Montel space by a closed subspace need not be semi-reflexive (IV, p. 63, exerc. 8).