Exercise 9, § 1

a) Let $E$ be a locally convex, Hausdorff, bornological and quasi-complete space. In order that the topologies $\tau(E', E)$ and $\sigma(E', E)$ on the dual $E'$ of $E$ be identical, it is necessary and sufficient that the topology of $E$ is the finest locally convex topology (show that every bounded subset of $E$ is finite dimensional). b) Let $E$ be a locally convex Hausdorff space, $E'$ its dual. In order that the strong topology $\beta(E', E)$ on $E'$ be identical with the weak topology $\sigma(E', E)$, it is necessary and sufficient that the topology of the bornological space associated with $E$ (III, p. 40, exerc. 1) is the finest locally convex topology on $E$.