The Krein–Milman Theorem and Its Applications

Abstract

One of the main merits of the functional analysis-based approach to problems of classical analysis is that it reduces problems formulated analytically to problems of a geometric character. The geometric objects that arise in this way lie in infinite-dimensional spaces, but they can be manipulated by using analogies with figures in the plane or in three-dimensional space. In the present chapter we add to the already built arsenal of geometric tools yet another one: the study of convex sets by means of their extreme points. We demonstrate Krein–Milman theorem on existence of extreme points in convex compact sets and give a number of applications, in particular the proof of the Stone–Weierstrass theorem invented by de Branges, Choquet’s proof of Bernstein’s representation for completely monotone functions, and Lindenstrauss’ proof of Lyapunov’s theorem on vector measures.