Exercise 1, § 6

Let $F, G$ be two vector spaces in duality, such that $\sigma(F, G)$ is Hausdorff. Show that if $\mathscr{T}$ is a Hausdorff topology compatible with the vector space structure of $F$ and coarser than $\sigma(F, G)$ (but not necessarily locally convex a priori), then $\mathscr{T} = \sigma(F, G_1)$, where $G_1$ is a vector subspace of G, dense in the topology $\sigma(G, F)$. (Consider on F the locally convex topology $\mathscr{T}_1$ in which a fundamental system of neighbourhoods of 0 is formed by the closed, convex, balanced sets in $\mathscr{T}$ which are neighbourhoods of 0 for $\sigma(F, G)$.) Deduce that if $\mathscr{T}_0$ is a Hausdorff locally convex topology on a vector space E, that is minimal in the set of Hausdorff locally convex topologies on E (II, p. 85, exerc. 13) it is also minimal in the set of topologies (locally convex or not) that are Hausdorff and compatible with the vector space structure of E.