Exercise 6, § 1

a) Let E be a locally convex Hausdorff space. For every ultrafilter $\mathscr{U}$ on E, let $\mathscr{U}'$ be the filter for which the convex envelopes of sets of $\mathscr{U}$ form a base. Show that, in order that a point of E be a limit of $\mathscr{U}$ for the weakened topology, it is necessary and sufficient that it is a limit point of $\mathscr{U}'$ for the initial topology (use exerc. 11 of II, p. 84). b) Let A be a convex subset of a vector space E and let $\mathscr{T}_1', \mathscr{T}_2'$ be two locally convex Hausdorff topologies on E, and $\mathscr{T}_1', \mathscr{T}_2'$ the corresponding weakened topologies. Show that if the topology induced on A by $\mathscr{T}_1'$ is finer than the topology induced by $\mathscr{T}_2'$, then the topology induced on A by $\mathscr{T}_1'$ is finer than the topology induced on A by $\mathscr{T}_2'$.