Exercise 1, § 1

a) Let X be a locally compact space. Show that every compact convex subset A of the locally convex space $\mathscr{K}(X; \mathbf{C})$ that is contained in $\mathscr{K}_+(X)$ is strictly compact. (Consider the upper envelope function of the set of all $f \in A,\ s = \sup_{f \in A} f$, which is continuous on X and tends to 0 at the point at infinity of X. For every continuous numerical function g on X, denote by P(g) the set of $x \in X$ such that $g(x) > 0$. Show that there exists a function f in A such that $P(f) = P(s)$; for this, observe that if $K_n$ is the compact set of $x \in X$ such that $s(x) \geq 1/n$, there exists a finite number of functions $f_{n,i} \in A$ such that the open sets $P(f_{n,i})$ cover $K_n$. Then make use of the fact that A is convex and closed.)

b) Assume that there exists in X a countable dense subset D. Show that every compact convex subset A of $\mathscr{K}(X; \mathbf{C})$ is then strictly compact. (Let U be the set of $x \in X$ such that there exists an $f \in A$ for which $f(x) \neq 0$, which is an open set in X. Show, making use of the fact that A is a Baire space, that there exists a function $g \in A$ such that $g(x) \neq 0$ for all $x \in U \cap D$.)