Lemma 1

The closure in ${F'}^*$ (for the topology $\sigma({F'}^*, F')$) of every bounded subset of F is contained in the bidual $F''$.

For, a bounded subset of F is contained in the polar (in $F''$) of a neighborhood of 0 in the strong dual $F'$ of F, hence is relatively compact in $F''$ for $\sigma(F'', F')$ (TVS, III, §3, No. 5, Prop. 7 and No. 4, Cor. 2 of Prop. 4); since $\sigma(F'', F')$ is induced by $\sigma({F'}^*, F')$, the lemma is proved.