Krein–Milman theorem

E506849

theorem

The Krein–Milman theorem is a fundamental result in functional analysis and convex geometry stating that a compact convex set in a locally convex topological vector space is the closed convex hull of its extreme points.

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All labels observed (1)

Label	Occurrences
Krein–Milman theorem canonical	2

Statements (41)

Predicate	Object
instanceOf	theorem ⓘ
appliesTo	compact convex sets ⓘ locally convex topological vector spaces ⓘ
assumption	The ambient space is a locally convex topological vector space. ⓘ The set is compact. ⓘ The set is convex. ⓘ
category	theorem about convex sets ⓘ theorem about topological vector spaces ⓘ
conclusion	A compact convex set equals the closed convex hull of its extreme points. ⓘ
coreIdea	Compact convex sets are generated by their extreme points via closed convex hull. ⓘ
doesNotRequire	finite dimensionality of the space ⓘ
field	convex geometry ⓘ functional analysis ⓘ
generalizes	finite-dimensional results about polytopes and extreme points ⓘ
hasConsequence	existence of extreme points in many optimization problems ⓘ structure theory of compact convex sets in locally convex spaces ⓘ
holdsIn	Hausdorff locally convex topological vector spaces ⓘ
implies	Every nonempty compact convex set in a locally convex space has at least one extreme point. ⓘ
involvesConcept	closed convex hull ⓘ compactness ⓘ convex hull ⓘ extreme point ⓘ local convexity ⓘ topological vector space ⓘ
isFundamentalResultIn	convex analysis ⓘ topological vector space theory ⓘ
namedAfter	David Milman NERFINISHED ⓘ Mark Krein NERFINISHED ⓘ
originalAuthors	David Milman NERFINISHED ⓘ Mark Krein NERFINISHED ⓘ
relatedTo	Bauer maximum principle NERFINISHED ⓘ Choquet theory NERFINISHED ⓘ Choquet–Bishop–de Leeuw theorem NERFINISHED ⓘ Minkowski theorem NERFINISHED ⓘ
requires	Hahn–Banach separation theorems in its proof ⓘ
statement	Every compact convex subset of a locally convex topological vector space is the closed convex hull of its extreme points. ⓘ
usedIn	duality theory in functional analysis ⓘ probability measures on compact convex sets ⓘ representation of points in convex sets by extreme points ⓘ study of state spaces in C*-algebras ⓘ
yearProved	1940 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Carathéodory’s theorem in convex geometry → relatedTo → Krein–Milman theorem ⓘ

Banach–Alaoglu theorem → relatedTo → Krein–Milman theorem ⓘ