Exercise 1, § 2

$a$) Let $E$ be a locally convex Hausdorff space, and $E'$ its dual. Show that the following properties are equivalent : $\alpha$ $E$ is infra-barrelled (III, p. 44, exerc. 7). $\beta$ Every strongly bounded subset in $E'$ is equicontinuous. $\gamma$ Every strongly bounded subset in $E'$ is relatively weakly compact, and the initial topology of $E$ is $\tau(E, E')$. $\delta$ The topology induced on $E$ by the strong topology of the bidual $E''$ is identical with the initial topology on $E$. Then a fundamental system of neighbourhoods of 0 for the strong topology of $E''$ consists of the closures, for the topology $\sigma(E'', E')$, of a fundamental system of neighbourhoods of 0 for the initial topology of $E$. b) Prove that if $E$ is infra-barrelled and if its dual $E'$ is identical with its algebraic dual $E^*$, then the initial topology of $E$ is the finest locally convex topology.