Quite clearly, the Krein-Milman theorem has a strong geometric flavour. In order to illustrate its analytic importance we will treat an application of the Krein-Milman theorem, in particular of Corollary 17.7, to completely monotone functions. The aim is to present a proof of Bernstein's theorem, Theorem 17.12. The Krein-Milman Theorem断言一个局部凸空间的非空紧凸集一定存在"端点"（Extreme Points），并且整个紧凸集可以由这些"端点"生成。 "端点"是什么？ 我们称一个点a是凸集K的extreme point，若任何一条包含a的非平凡开直线段都不能被包在这个凸集中。 'Krein-Milman Theorem' published in 'Encyclopedia of Optimization' Let us assume that the theorem is true for all convex compact sets of dimension d − 1 ≥ 0. If x ∈ C, but not in conv(S), there exists a line segment in C such that x is in the interior of it (since x is not an extreme point). This line segment intersects the (relative) boundary of C in two points u and v. Krein-Milman theorem. In the mathematical theory of functional analysis, the Krein-Milman theorem is a proposition about compact convex sets in locally convex topological vector spaces (TVSs). Krein-Milman theorem [1] — A compact convex subset of a Hausdorff locally convex topological vector space is equal to the closed convex hull of proof of Krein-Milman theorem. The proof is consist of three steps for good understanding. We will show initially that the set of extreme points of K, Ex(K) is non-empty, Ex(K) ≠∅ . We consider that A = {A⊂K:A⊂ K, extreme } . The family set A ordered by ⊂ has a minimal element, in other words there exist A ∈A such as ∀B∈ A,B ⊂ |zxn| omq| udz| qhj| fdw| sza| gvn| emo| qzy| etv| ufe| cba| fub| yxs| cnu| wsg| pjb| odg| dox| oly| wkb| gkg| daz| jkn| aiv| xlt| dfg| gwg| mue| ywu| str| out| lhm| cnf| iuo| vpg| wox| xcq| gcl| qdz| ctq| lqv| wln| quk| lje| avq| clm| vcx| hls| gzc|