Remark

Let $E$ be a locally convex Hausdorff space, $E'$ its dual and $E''$ its bidual. We have $E \subset E'' \subset {E'}^*$, where ${E'}^*$ is the algebraic dual of $E'$. If $B$ is a bounded subset of $E$, its closure $\overline{B}$ in ${E'}^*$ endowed with $\sigma({E'}^*, E')$ is contained in $E''$: for, the polar $U = B^\circ$ of $B$ in $E'$ is a neighbourhood of 0 in $E'_b$, and we have $\overline{B} \subset U^\circ \subset E''$.