Exercise 10, § 3

Let E be a Fréchet space satisfying the first axiom of countability. Show that if in the dual $E'$ of E, every sequence which converges for the weak topology $\sigma(E', E)$ also converges for the strong topology $\beta(E', E)$, then E is a Montel space. (Show that every bounded subset of $E'$ is relatively compact for the strong topology; use GT, II, § 4, exerc. 6; then use IV, p. 53, exerc. 11, b).)