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, , stating that S is the inverse limit of a sequence of finite dimensional simplices, and the idea used in Example 2.6 of  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].