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I want to show the weak star closed convex hull of a finite set of points is contained in the linear span of those points.

It's enough to show that any finite dimensional subspace $V$ of a Banach space $Z$ is weak star closed in $Z$. Since $V$ is finite dimensional, it is a closed subspace of $Z$ in the norm topology. How can I show that it is also weak star closed? Also, it this result true for arbitrary subspaces?

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  • $\begingroup$ Is $Z$ given as the dual space of another Banach space? Otherwise what do you mean by weak star? $\endgroup$ – Robert Israel Aug 6 '18 at 22:48
  • $\begingroup$ Yes, if $Z=X^*$ for some Banach space $X$ $\endgroup$ – Tom Chalmer Aug 7 '18 at 21:27
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All finite diemensional subspaces of Hausdorff topological vector spaces are closed. This is theorem 1.21 in Rudin's Functional Analysis.

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This is true not only for arbitrary subspaces of Banach spaces, but in fact for arbitrary subspaces of locally convex spaces. For any locally convex space $X$, the weak and original closures of any convex set $E \subset X$ are the same (see, for instance, Theorem 3.12 here). Since Banach spaces are locally convex and subspaces of any space are convex, this shows that the weak and original closures of any subspace of a Banach space are the same.

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    $\begingroup$ If the OP meant weak-* (presumably relative to a Banach space $X$ such that $Z = X^*$), this is not relevant. $\endgroup$ – Robert Israel Aug 6 '18 at 22:50

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