Banach–Stone theorem

E421067

mathematical theorem theorem in functional analysis

The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.

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All labels observed (3)

Label	Occurrences
Banach–Stone theorem canonical	1
complex Banach–Stone theorem	1
real Banach–Stone theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T4219687 — 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: Banach–Stone theorem
Context triple: [Stefan Banach, eponymOf, Banach–Stone theorem]

A. 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.
B. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
C. 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.
D. Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
E. Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² 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: Banach–Stone theorem
Target entity description: The Banach–Stone theorem is a fundamental result in functional analysis that characterizes compact Hausdorff spaces via isometric isomorphisms between their spaces of continuous real- or complex-valued functions.

A. 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.
B. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
C. 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.
D. Banach inverse mapping theorem
The Banach inverse mapping theorem is a fundamental result in functional analysis stating that a bijective bounded linear operator between Banach spaces has a bounded linear inverse.
E. Riesz–Fischer theorem
The Riesz–Fischer theorem is a fundamental result in functional analysis that establishes the equivalence between square-summable sequences and square-integrable functions, providing the foundation for the Hilbert space structure of L² spaces.
F. None of above. chosen

Statements (44)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in functional analysis ⓘ
appliesTo	C(X) ⓘ C(Y) ⓘ
assumes	surjective linear isometry between C(X) and C(Y) ⓘ
characterizes	compact Hausdorff spaces up to homeomorphism ⓘ isometric isomorphisms between C(X) and C(Y) ⓘ
codomainCondition	C(X) Banach space with sup norm ⓘ C(Y) Banach space with sup norm ⓘ
concerns	compact Hausdorff spaces ⓘ isometric isomorphisms of Banach spaces ⓘ spaces of continuous functions ⓘ
concludes	existence of homeomorphism φ:Y→X ⓘ representation of isometry via composition with φ and multiplication by unimodular function ⓘ
domainCondition	X compact Hausdorff space ⓘ Y compact Hausdorff space ⓘ
field	functional analysis ⓘ topology ⓘ
hasGeneralization	Banach–Stone type theorems for C0(X) ⓘ Banach–Stone type theorems for Lipschitz function spaces ⓘ Banach–Stone type theorems for vector-valued function spaces ⓘ Gelfand–Naimark theorem ⓘ
hasVariant	complex Banach–Stone theorem ⓘ linked to: Banach–Stone theorem real Banach–Stone theorem ⓘ linked to: Banach–Stone theorem
holdsFor	continuous complex-valued functions on compact Hausdorff spaces ⓘ continuous real-valued functions on compact Hausdorff spaces ⓘ
implies	isometric isomorphism class of C(X) determines X up to homeomorphism ⓘ topological structure of X is determined by Banach space structure of C(X) ⓘ
mathematicsSubjectClassification	46E15 ⓘ 54C35 ⓘ
namedAfter	Marshall Harvey Stone ⓘ linked to: Marshall H. Stone Stefan Banach ⓘ
relatedTo	C(K) spaces ⓘ Gelfand representation of commutative C*-algebras ⓘ Riesz representation theorem ⓘ
statement	Every surjective linear isometry T:C(X)→C(Y) is induced by a homeomorphism between X and Y and a unimodular function ⓘ If C(X) and C(Y) are isometrically isomorphic as Banach spaces then X and Y are homeomorphic ⓘ
topic	duality between topology and function spaces ⓘ isometric classification of C(K) spaces ⓘ
uses	homeomorphism ⓘ linear isometry ⓘ supremum norm ⓘ
valueType	complex-valued continuous functions ⓘ real-valued continuous functions ⓘ

How these facts were elicited

Referenced by (3)

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

Stefan Banach → eponymOf → Banach–Stone theorem ⓘ

Banach–Stone theorem → hasVariant → real Banach–Stone theorem ⓘ

linked to: Banach–Stone theorem

Banach–Stone theorem → hasVariant → complex Banach–Stone theorem ⓘ