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In the mathematical theory of functional analysis, the Krein–Milman theorem is a proposition about convex sets in topological vector spaces. A particular case of this theorem, which can be easily visualized, states that given a convex polygon, one only needs the corners of the polygon to recover the polygon shape. The statement of the theorem is false if the polygon is not convex, as then there can be many ways of drawing a polygon having given points as corners. Formally, let be a locally convex topological vector space (assumed to be Hausdorff), and let be a compact convex subset of . Then, the theorem states that is the closed convex hull of its extreme points. The closed convex hull above is defined as the intersection of all closed convex subsets of that contain This turns out to be the same as the closure of the convex hull in the topological vector space. One direction in the theorem is easy; the main burden is to show that there are 'enough' extreme points. The original statement proved by Mark Krein and David Milman was somewhat less general than this. Hermann Minkowski had already proved that if is finite-dimensional then equals the convex hull of the set of its extreme points. The Krein–Milman theorem generalizes this to arbitrary locally-convex , with a caveat: the closure may be needed. ==Relation to the axiom of choice== The axiom of choice, or some weaker version of it, is needed to prove this theorem in Zermelo–Fraenkel set theory. This theorem together with the Boolean prime ideal theorem, though, can prove the axiom of choice. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Krein–Milman theorem」の詳細全文を読む スポンサード リンク
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