Krein–Milman Theorem

What is 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 — A compact convex subset of a Hausdorff locally convex topological vector space is equal to the closed convex hull of its extreme points. This theorem generalizes to infinite-dimensional spaces and to arbitrary compact convex sets the following basic observation: a convex (i.e. "filled") triangle, including its perimeter and the area "inside of it", is equal to the convex hull of its three vertices, where these vertices are exactly the extreme points of this shape. This observation also holds for any other convex polygon in the plane ℝ2.

Technology Types

convex hullmathematical theoremoriented matroidpropositionspacestatementtheoremtheorems in convex geometrytheorems in discrete geometrytheorems in functional analysistheorems in geometrytheorytopological vector space

Synonyms

Krein-Mil'man theoremKrein-MilmanKrein-Milman theoremKrein–Mil'man theorem

Translations

Satz von Krein-Milman (de)Teorema di Krein-Milman (it)Théorème de Krein-Milman (fr)Twierdzenie Krejna-Milmana (pl)Теорема Крейна — Мильмана (ru)크레인-밀만 정리 (ko)クレイン＝ミルマンの定理 (ja)

Tech Info

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— Date merged: 11/6/2021, 1:32:49 PM
— Date scraped: 5/20/2021, 5:54:23 PM