Exercise 10, § 3

Let $E$ be a Fréchet space of countable type. Now prove that if, in the dual $E'$ of $E$, every sequence convergent for the weak topology $\sigma(E', E)$ is also convergent for the strong topology $\beta(E', E)$, then $E$ is a Montel space. (Now prove that every bounded subset of $E'$ is relatively compact for the strong topology, using TG, II, p. 37, exerc. 6; then use IV, p. 54, exerc. 11, b).)