Exercise 6, § 5

a) Let X be a compact space, H a convex subset of the product space $\mathbf{R}^X$, consisting of continuous functions on X. Suppose that every decreasing sequence of non-empty convex and closed subsets in H has a non-empty intersection. Show that the closure $\overline{H}$ of H in $\mathbf{R}^X$ is compact and consists of continuous functions on X. (Argue by reductio ad absurdum : consider a non continuous function $u \in \overline{H}$; show that then there will exist a point $a \in X$, a number $\delta > 0$, a sequence $(x_n)$ points of X and a sequence $(f_m)$ of functions in H such that : $1^\circ\ |u(x_n) - u(a)| \geq \delta$ for all $n$; $2^\circ\ |f_m(x_n) - f_m(a)| \leq \frac{\delta}{8}$ for $m \leq n$; $3^\circ\ |u(x_n) - f_m(x_n)| \leq \frac{\delta}{8}$ and $|u(a) - f_m(a)| \leq \frac{\delta}{8}$ for $m \geq n + 1$. Consider a limit point b of the sequence $(x_n)$, and a function $f$ belonging to the intersection of the $A_m$, where $A_m$ is the closed convex envelope in H of the set of all $f_k$ for $k \geq m.$ b) Let E be a Hausdorff and quasi-complete locally convex space, $E'$ its dual. Let H be a convex subset of E such that every decreasing sequence of non-empty and closed convex subsets of H has a non-empty intersection; show that H is relatively compact in E for the topology $\sigma(E, E')$. (Reduce to the case where E is complete; consider E to be embedded in ${E'}^*$ and use a) and also III, p. 21, cor. 1.)