Exercise 2, § 1

a) Let F and G be two vector spaces in separating duality. Show that, if F is relatively bounded for $\sigma(F, G)$ (III, p. 43, exerc. 6), then G is relatively bounded for $\sigma(G, F)$.

b) Let F be a vector space, and let G₁, G₂ be two vector subspaces of F* such that F is in separating duality with G₁ and with G₂. Show that, if F is relatively bounded for σ(F, G₁) and σ(F, G₂), it is so also for σ(F, G₁ + G₂). c) Suppose that F has a countable basis. Show that, for every vector subspace G of F* which is in separating duality with F and has a countable basis, F is relatively bounded for σ(F, G) (by induction define two bases (aₙ), (bₙ) of F and G respectively, such that $\langle a_m, b_n \rangle = \delta_{mn}$).