Exercise 5, § 4

Let E and F be two normed spaces, u a continuous linear mapping from E into F. a) Show that if u is a strict morphism from E into F, then 'u is a strict morphism from the strong dual F'_b into the strong dual E'_b. b) Suppose E is complete; show that if 'u is a strict morphism from F'_b into E'_b, then u is a strict morphism from E into F, and 'u is a strict morphism from F' into E' for the weak topologies σ(F', F) and σ(E', E) (consider F as a subspace of its completion). c) Give an example where E is not complete, F is complete, 'u is a strict injective morphism from F' into E' for the strong topologies and for the weak topologies, but u is not a strict morphism from E into F (cf. II, p. 74, exerc. 5). d) Give an example where E is not complete, F is complete, u is a strict injective morphism from E into F, 'u is a strict morphism from F' into E' for the strong topologies, but not for the weak topologies (take E to be everywhere dense in F). e) If F is complete and if 'u is a strict morphism from F' into E' for the weak topologies, then 'u is a strict morphism from F' into E' for the strong topologies (extend u to Ē). f) Give an example where E is complete, F is not complete, 'u is a strict morphism from F' into E' for the weak topologies but not for the strong topologies (cf. II, p. 74, exerc. 5).