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I wonder if Hahn Banach theorem still holds on a weak* closed set.

To be specific, let $X$ be a Banach space, and $X^*$ is its dual space. Let $S$ be a convex, weak* closed set in $X^*$, and $f\in X^*\backslash S$. Then, is there any point $y\in X$ such that $f(y) > g(y)$ for any $g\in S$?

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    $\begingroup$ The Hahn-Banach separation theorem applies to locally convex spaces, so yes what you've stated is true. $\endgroup$ – Aweygan Jun 6 '17 at 23:36
  • $\begingroup$ ... because the dual of $X^\ast$ endowed with the weak$^\ast$ topology is $X$. $\endgroup$ – Jochen Jun 8 '17 at 7:41

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