Exercise 10, § 1

Let $E$ be a locally convex Hausdorff space, $E'$ its dual. Show that the following propositions are equivalent : $\alpha)$ $E$ is barrelled; $\beta)$ every weakly bounded subset of $E'$ is equicontinuous; $\gamma)$ every weakly bounded subset of $E'$ is relatively weakly compact, and the topology of $E$ is $\tau(E, E')$.