Exercise 20, § 1

Let E be a Hausdorff locally convex space, and $E'$ its dual. a) Let M be a closed vector subspace of E. The topology $\tau(M, E'/M^\circ)$ is identical with the topology induced on M by $\tau(E, E')$ if and only if every convex balanced set in $E'/M^\circ$ which is compact for the weak topology $\sigma(E'/M^\circ, M)$, is the canonical image of a convex balanced, compact (for $\sigma(E', E)$) subset of $E'$ (cf. V, p. 73, exerc. 15). b) Let N be a dense vector subspace of E. If the topology induced on N by that of E is identical with $\tau(N, E')$, show that the topology of E is identical with $\tau(E, E')$.