Exercise 8, § 3

Let E be a Fréchet space, $E'_b$ its strong dual. Suppose that there exists an everywhere dense sequence $(x'_n)$ in $E'_b$. Show that E satisfies the first axiom of countability. (Let $(K'_n)$ be a countable base for the canonical bornology of $E'_b$ consisting of closed convex balanced sets. For every system $\alpha$ consisting of a point $x'_n$, an arbitrary finite number of rational numbers $\lambda_k > 0 (1 \leq k \leq m)$ and m indices $n_k$ such that $x'_n \notin 2 \sum_{k=1}^m \lambda_k K'_{n_k} = 2H'_\alpha$, let $x_\alpha \in E$ be such that the hyperplane with equation $\langle x_\alpha, y' \rangle = 1$ strictly separates the two weakly compact sets $H'_\alpha$ and $x'_n + H'_\alpha$. Show that for every $x' \neq 0$ in $E'$, there exists a system $\alpha$ such that $\langle x_\alpha, x' \rangle \neq 0$. For this, consider a neighbourhood V' of 0 in $E'_b$ such that $V' \cap (x' + V') = \varnothing$, then for each integer $m$, take a rational number $\lambda_m > 0$ such that $\lambda_m K'_m \subset V'$; use the fact that the union $U' \subset V'$ of the $\lambda_m K'_m$ is a neighbourhood of 0 (exerc. 7, and IV, p. 58, exerc. 3, b)) and that there exists $n$ such that $x'_n \in x' + U'$.)