# Krein-Milman theorem

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## Krein-Milman theorem

[′krīn ′mil·mən ‚thir·əm]
(mathematics)
The theorem that in a locally convex topological vector space, any compact convex set K is identical with the intersection of all convex sets containing the extreme points of K.
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This result is an extension of the classical Krein-Milman Theorem.
As mentioned in the introduction, several results have been extended to the hyperspace WCC (X): In the next section, we shall further extend the notion of hyperspace and its corresponding topology [T.sub.w] where the underlying space X is a locally convex topological vector space instead of a Banach space and prove an extreme point theorem which is an extension of the classical Krein-Milman Theorem.
Given a Banach space E, let [B.sub.E**] denote the unit closed ball of the second dual E**; and let Ext([B.sub.E**]) be the set of all extreme points of [B.sub.E**] (which is of course nonvoid by virtue of the Krein-Milman theorem).
The collection concludes with papers on the theory of complex functions, a proof of the Krein-Milman Theorem, and a review of the influence of Wedderburn on modern algebra.
The proof of this relies on a Krein-Milman theorem for Markov operators obtained in Section 2, a result of Goodearl, [6], stating that S is the inverse limit of a sequence of finite dimensional simplices, and the idea used in Example 2.6 of [9] which exchanges an arbitrary inductive limit C*-algebra of the above type with one which is simple and has the same tracial state space.
By the Hahn-Banach and the Krein-Milman Theorems, E(x) [not equal to] [empty set].

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