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- Krein–Milman_theorem abstract "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.".
- Krein–Milman_theorem thumbnail Extreme_points_illustration.png?width=300.
- Krein–Milman_theorem wikiPageID "3206654".
- Krein–Milman_theorem wikiPageRevisionID "595016005".
- Krein–Milman_theorem id "5921".
- Krein–Milman_theorem title "Krein–Milman theorem".
- Krein–Milman_theorem subject Category:Convex_hulls.
- Krein–Milman_theorem subject Category:Oriented_matroids.
- Krein–Milman_theorem subject Category:Theorems_in_convex_geometry.
- Krein–Milman_theorem subject Category:Theorems_in_discrete_geometry.
- Krein–Milman_theorem subject Category:Theorems_in_functional_analysis.
- Krein–Milman_theorem subject Category:Topological_vector_spaces.
- Krein–Milman_theorem comment "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.".
- Krein–Milman_theorem label "Krein–Milman theorem".
- Krein–Milman_theorem label "Satz von Krein-Milman".
- Krein–Milman_theorem label "Teorema di Krein-Milman".
- Krein–Milman_theorem label "Théorème de Krein-Milman".
- Krein–Milman_theorem label "Twierdzenie Kreina-Milmana".
- Krein–Milman_theorem label "Теорема Крейна — Мильмана".
- Krein–Milman_theorem sameAs Krein%E2%80%93Milman_theorem.
- Krein–Milman_theorem sameAs Satz_von_Krein-Milman.
- Krein–Milman_theorem sameAs Théorème_de_Krein-Milman.
- Krein–Milman_theorem sameAs Teorema_di_Krein-Milman.
- Krein–Milman_theorem sameAs 크레인-밀만_정리.
- Krein–Milman_theorem sameAs Twierdzenie_Kreina-Milmana.
- Krein–Milman_theorem sameAs Q642620.
- Krein–Milman_theorem sameAs Q642620.
- Krein–Milman_theorem wasDerivedFrom Krein–Milman_theorem?oldid=595016005.
- Krein–Milman_theorem depiction Extreme_points_illustration.png.