Theorem 17.1.6 (Goldstine).label Let $\dpn{E, F}{\lambda}$ be a duality over $K \in \RC$, $A \subset E$ be non-empty, convex, circled, and $\sigma(E, F)$-compact, and $B$ be the closed unit ball of $E_{A}^{*}$, then $B \cap F$ is $\sigma(E_{A}^{*}, E_{A})$-dense in $B$.
Proof. For any $S \subset E$ or $S \subset F$, denote $S^{\circ}$ as the polar of $S$ with respect to $\dpn{E, F}{\lambda}$. For any $S \subset E_{A}$ or $S \subset E_{A}^{*}$, denote $S^{\bullet}$ as the polar of $S$ with respect to $\dpn{E_A, E_A^*}{E_A}$.
Since $B_{E_A}(0, 1)$ is circled and $A = \ol{B_{E_A}(0, 1)}^{E_A}$ is compact in $E$,
is the polar of $A$ with respect to $\dpn{E, F}{\lambda}$. Now, as $A$ is convex, circled, and compact, the Bipolar Theorem implies that
Given that $B \cap F$ is a convex and circled subset of $E_{A}^{*}$,
$\square$
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