Exercise 7, § 5

Let E be a Fréchet space satisfying the first axiom of countability, $E'$ its dual. Show that for a convex subset $A'$ of $E'$ to be closed for $\sigma(E', E)$, it is sufficient that, if a sequence $(x'_n)$ of points of $A'$ has a limit $a'$ in $E'$ for $\sigma(E', E)$, then $a' \in A'$.