Corollary 2

Let $E$ be a Fréchet space. In order that a convex subset $A$ of the dual $E'$ of $E$ be closed for $\sigma(E', E)$, it is necessary and sufficient that $A \cap U^{\circ}$ is closed for $\sigma(E', E)$ for every neighbourhood $U$ of 0 in $E$.

Since $E$ is complete, the topology $\mathscr{T}_c$ on $E'$ is compatible with the duality between $E'$ and $E$ (IV, p. 3, Example); consequently the closed convex subsets in $E'$ are the same for $\mathscr{T}_c$ and $\sigma(E', E)$ (IV, p. 1, prop. 1). The corollary then follows from the identity of the topologies $\mathscr{T}_c$ and $\mathscr{T}_f$.

Recall (I, p. 13) that the hyperplanes of $E'$ which are closed for $\sigma(E', E)$ are the kernels of linear forms on $E'$ associated with elements of $E$. Cor. 2 therefore gives another proof (for Fréchet spaces) of cor. 1 of III, p. 21.