Exercise 4, § 3

. β) $E'_b$ is infra-barrelled (III, p. 45, exer. 7). γ) $E'_b$ is bornological. δ) $E'_b$ is barrelled. ε) $E'_b$ is ultrabornological (III, p. 47, exer. 19). c) Now prove that if $E'_b$ is reflexive, one has $\hat{E} = E''$ (which is obviously reflexive) (cf. IV, p. 52, exer. 4 and p. 54, exer. 11).

4) Let E be a separated locally convex space, $E'$ its dual. If M is a closed, metrizable and distinguished vector subspace of E (IV, p. 52, exer. 4), the strong topology $\beta(E'/M^\circ, M)$ is the quotient topology by $M^\circ$ of the strong topology $\beta(E', E)$ (use exer. 3, b) and IV, p. 51, exer. 22, b)).