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Let $X$ be a Banach space. Consider the dual $X^*$ equipped with the weak*-topology.

Is it true that the closed convex hull of a compact subset $K$ of the dual $X^*$ is compact?

ps: I know that the closed convex hull of a compact subset of a Banach space is compact.

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  • $\begingroup$ I believe (but this needs to be checked) that weak*-compact sets are norm-bounded. If true, that would mean $K\subseteq nB_{X^*}$ for some $n\in\mathbb{N}$. Now $\overline{\text{co}K}^{w*}\subseteq nB_{X^*}$ by the convexity of $B_{X^*}$ together with Banach-Alaoglu. $\endgroup$ – Ben W Jun 11 '16 at 15:56

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