Corollary 17.1.7.label Let $E$ be a normed space over $K \in \RC$, then $E \cap \ol{B_{E^{**}}(0, 1)}$ is $\sigma(E^{**}, E^{*})$-dense in $\ol{B_{E^{**}}(0, 1)}$.
Proof. By the Banach-Alaoglu Theorem, $\ol{B_{E^*}(0, 1)}$ is convex, circled, and $\sigma(E^{*}, E)$-compact. By Goldstine’s Theorem, $E \cap \ol{B_{E^{**}}(0, 1)}$ is $\sigma(E^{**}, E^{*})$-dense in $\ol{B_{E^{**}}(0, 1)}$.$\square$
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