Exercise 11, § 6

Let F, G be two vector spaces in separating duality. An ultrafilter $\mathscr{U}$ on F converges weakly to a point $x_0$ if, and only if, $x_0$ belongs to the intersection of all the weakly closed convex sets which belong to $\mathscr{U}$ (note that if $x_0$ is a point of this intersection that is not a cluster point of $\mathscr{U}$, then there exists a closed half-space belonging to $\mathscr{U}$ and not containing $x_0$). Deduce from this result that, for a sequence of points $(x_n)$ of F to be weakly convergent to a point $a$, it is necessary and sufficient that $a$ belongs to all the weakly closed convex envelopes of the sets formed by an infinity of the terms of the sequence (use prop. 7 of GT, I, § 6 . 4).