proof of Krein-Milman theorem

The proof is consist of three steps for good understanding. We will show initially that the set of extreme pointsPlanetmathPlanetmath of K,Ex(K) is non-empty, Ex(K). We consider that 𝒜={AK:AK,extreme}.
The family set 𝒜 ordered by has a minimal element, in other words there exist A𝒜 such as B𝒜,BA we have that B=A.
We consider A<BBA,A,B𝒜. The ordering relation < is a partially relation on 𝒜. We must show that A is maximal element for 𝒜. We apply Zorn’s lemma.We suppose that


is a chain of 𝒜.Witout loss of generality we take A=iIAi and then A. 𝒞 has the property of finite intersectionsMathworldPlanetmath and it is consist of closed sets. So we have that iIAi. It is easy to see that A𝒜. Also AAi, for any iI, so we have that A>Ai, for any iI.
Every minimal element of 𝒜 is a set which has only one point.
We suppose that there exist a minimal element A of 𝒜 which has at least two points, x,yA. There exist x*X* such as x*(x)x*(y), witout loss of generality we have that x*(x)<x*(y). A is compact set (closed subset of the compact K). Also there exist α such that α=supzAx*(z) and B={zA:x*(z)=α}. It is obvious that B is an extreme subset of A, B is an extreme subset of K,B𝒜. xB since B𝒜 and BA that contradicts to the fact that A is minimalPlanetmathPlanetmath extreme subset of 𝒜.
From the above two steps we have that Ex(K).
K=c¯o(Ex(K)) where c¯o(Ex(K)) denotes the closed convex hullMathworldPlanetmath of extreme points of K.
Let L=c¯o(Ex(K)). Then L is closed subset of K, therefore it is compact, and convex clearly by the definition. We suppose that LK. Then there exist xK-L. Let use Hahn-Banach theoremMathworldPlanetmath(geometric form). There exist x*X* such as supwLx*(w)<x*(x). Let α=sup{x*(y):yK}, B={yK:x*(y)=α}. SimilarMathworldPlanetmathPlanetmath to Step2 B is extreme subset of K. B is compact and from step1 and step2 we have that Ex(B). It is true that Ex(B)Ex(K)L. Now let yEx(B) then x*(y)=α and if yLx*(y)<x*(x)α. That is a contradictionMathworldPlanetmathPlanetmath.

Title proof of Krein-Milman theorem
Canonical name ProofOfKreinMilmanTheorem
Date of creation 2013-03-22 15:24:40
Last modified on 2013-03-22 15:24:40
Owner georgiosl (7242)
Last modified by georgiosl (7242)
Numerical id 6
Author georgiosl (7242)
Entry type Proof
Classification msc 46A03
Classification msc 52A07
Classification msc 52A99