Exercise 2, § 4

Let E be a locally convex Hausdorff space such that in the dual E' there exists an infinite dimensional convex set B' which is compact for σ(E', E) (this condition is realized for example, when E is an infinite dimensional vector space, endowed with σ(E, E*)). Show that there exists a linear form u ∈ E'* which is unbounded on B'; deduce that B' is not compact for the topology σ(E', F), where F is the subspace E + Ru of E'*. Conclude from this that the canonical injection from E into F is a strict morphism for the topologies σ(E, E') and σ(F, E'), but not for the topologies τ(E, E') and τ(F, E').