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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Label	Occurrences
Krein–Milman theorem canonical	3

How this entity was disambiguated

This entity first appeared as the object of triple T5256295 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Krein–Milman theorem
Context triple: [Carathéodory’s theorem in convex geometry, relatedTo, Krein–Milman theorem]

A. Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
B. Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
C. Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
D. Banach–Saks theorem
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
E. Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Krein–Milman theorem
Target entity description: 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.

A. Banach–Alaoglu theorem
The Banach–Alaoglu theorem is a fundamental result in functional analysis stating that the closed unit ball in the dual of a normed space is compact in the weak-* topology.
B. Hahn–Banach theorem
The Hahn–Banach theorem is a fundamental result in functional analysis that guarantees the extension of bounded linear functionals from a subspace to the whole space without increasing their norm.
C. Banach–Mazur theorem
The Banach–Mazur theorem is a fundamental result in functional analysis that characterizes separable Banach spaces as isometrically isomorphic to closed subspaces of spaces of continuous functions on compact metric spaces.
D. Banach–Saks theorem
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
E. Banach–Steinhaus theorem
The Banach–Steinhaus theorem is a fundamental result in functional analysis that characterizes when a family of continuous linear operators is uniformly bounded, with major implications for the behavior of sequences of operators on Banach spaces.
F. None of above. chosen

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 ⓘ Mark Krein ⓘ
originalAuthors	David Milman ⓘ Mark Krein ⓘ
relatedTo	Bauer maximum principle ⓘ Choquet theory ⓘ Choquet–Bishop–de Leeuw theorem ⓘ Minkowski theorem ⓘ linked to: Minkowski’s theorem on convex sets
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 ⓘ

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Referenced by (3)

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 ⓘ

Blaschke selection theorem → relatedTo → Krein–Milman theorem ⓘ

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Statements (41)

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Referenced by (3)

How this entity was disambiguated

How these facts were elicited