Exercise 12, § 4

Let E be a vector space with an enumerable basis $(e_n)$. Let V be the balanced convex envelope of the set of the $e_n$ and let W be the balanced convex envelope of the set of points

Let $\mathscr{T}_1$ (resp. $\mathscr{T}_2$) be the locally convex topology on E for which a fundamental system of neighbourhoods of 0 is formed by the $\lambda V$ (resp. $\lambda W$) for $\lambda > 0$. Show that $\mathscr{T}_1$ and $\mathscr{T}_2$ are Hausdorff, but that the lower bound of $\mathscr{T}_1$ and $\mathscr{T}_2$ in the set of locally convex topologies on E is not Hausdorff (cf. II, p. 80, exerc. 26).