Exercise 4, § 4

Let E and F be two locally convex Hausdorff spaces, u a continuous linear mapping from E into F, and M an everywhere dense vector subspace of E. Show that if the restriction of u to M is a strict morphism from M into F, then u is a strict morphism from E into F (use prop. 2 of IV, p. 27). If in addition u(M) = F, show that for every open, convex and balanced neighbourhood V of 0 in E, u(V) is in the interior of u(V ∩ M).