Exercise 3, § 3

Let E and F be two Hausdorff topological vector spaces over a non-discrete valued division ring; and suppose that E is metrisable and complete. Let $u$ be a continuous linear injection of E in F, and let G be a vector subspace of $u(E)$; suppose that there exists on G a topology $\mathscr{T}$ which is finer than the topology induced by that of F, is compatible with the vector space structure of G and for which G is metrisable and complete. Show that the mapping inverse to $u$, restricted to G, is continuous for $\mathscr{T}$ (use I, p. 19, cor. 5).