Let $X$ be a normed linear space and $K$ a bounded convex weak-$*$ closed subset of $X^{*}$. I need to show that $K$ possesses an extreme point; however, I am not entirely sure how to do this.

I suppose I could use Alaoglu's Theorem (Let $X$ be a normed linear spsace. Then the closed unit ball $B^{*}$ of its dual space is compact with respect to the weak $*$-topology) or one of its corollaries and the Krein-Milman Theorem (Let $K$ be a nonempty, compact, convex subset of a locally convex topological vector space $X$. Then, $K$ is the closed convex hull of its extreme points).

But, I'm not sure how to put these things together, and am honestly not that comfortable with dual spaces.

If someone could please help me put these things together and make some sense out of this, I would be very much appreciative.

Thank you.


1 Answer 1


Being bounded, $K$ is contained in some closed ball. A closed ball is compact in weak* topology (Alaoglu). In any topological space, a closed subset of a compact set is compact. Hence, $K$ is weak* compact.

It remains to apply Krein-Milman's theorem to $K$.


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