Exercise 5, § 5

a) Let X be a compact space, H an arbitrary subset of the space $\mathscr{C}(X)$ of continuous numerical functions on X. Let $f_0$ be a point of $\mathscr{C}(X)$ which is in the closure of H for the topology $\mathscr{T}_s$ of simple convergence on $\mathscr{C}(X)$. Show that there exists a countable subset $H_0$ of H such that $f_0$ is in the closure of $H_0$ for $\mathscr{T}_s$. (Show that for every pair of integers $m > 0, n > 0$ there exists a finite subset $H(m, n)$ of H with the following property : for every set of m points $t_k$ in X $(1 \leq k \leq m)$, there exists $f \in H(m, n)$ such that $|f_0(t_k) - f(t_k)| \leq 1/n$ for $1 \leq k \leq m.$) b) Let E be a locally convex metrizable space, $E'$ its dual and H a subset of E. Show that if $x_0$ is in the closure of H for the weakened topology $\sigma(E, E')$, then there exists a countable subset $H_0$ of H such that $x_0$ is in the closure of $H_0$ for this topology. (Use a), observing that $E'$ is the union of a countable family of compact sets for $\sigma(E', E').$