Exercise 2, § 3

Let E and F be two complete, metrisable vector spaces over a non-discrete valued division ring, and let $\mathscr{T}_0$ be the topology of F. Let $\mathscr{T}$ be a Hausdorff topology on F, coarser than $\mathscr{T}_0$. Show that if the linear mapping $u$ of E in F is continuous for the topology $\mathscr{T}$ on F, it is still continuous for the topology $\mathscr{T}_0$ on F (use the cor. 5 of I, p. 19). Deduce that if $\mathscr{T}_1$ and $\mathscr{T}_2$ are two distinct topologies on a vector space E over a non-discrete valued division ring, compatible with the vector space structure of E, and for each of them E is metrisable and complete, then there does not exist a Hausdorff topology on E coarser than $\mathscr{T}_1$ and $\mathscr{T}_2$. Give an example of two such topologies on an infinite dimensional vector space E (note that there exist bijections of E on itself such that both the bijection and its inverse are not continuous for a normed space topology on E).