Exercise 1, § 4

Let E and F be two locally convex Hausdorff spaces, E' and F' their respective duals. Show that if, for every vector subspace N of F, the topology induced on N by τ(F, F') is identical with τ(N, F'/N°) (IV, p. 51, exerc. 20), then every strict morphism from E into F for the topologies σ(E, E') and σ(F, F') is also a strict morphism for the topologies τ(E, E') and τ(F, F') (cf. IV, p. 10, prop. 11). Consider the case when F is metrizable.