Exercise 18, § 1

Let E, F be two vector spaces and E*, F* their respective algebraic duals. Show that if u is a linear mapping from F* into E, which is continuous for the topologies $\sigma(F^*, F)$ and $\sigma(E, E^*)$, then $u(F^*)$ is finite dimensional. (Use prop. 2 of IV, p. 27 to show that u is a strict morphism, and deduce that $u(F^*)$ is a subspace of E of minimal type (II, p. 85, exerc. 13); conclude by considering the bounded sets of this subspace.)