Stone–Weierstrass theorem

E480871

approximation theorem theorem in functional analysis

The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.

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All labels observed (3)

Label	Occurrences
Stone–Weierstrass theorem canonical	3
complex Stone–Weierstrass theorem	1
real Stone–Weierstrass theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T4927201 — 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: Stone–Weierstrass theorem
Context triple: [Weierstrass approximation theorem, generalizationOf, Stone–Weierstrass theorem]

A. Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
B. Banach–Stone theorem
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.
C. 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.
D. 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.
E. 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.
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: Stone–Weierstrass theorem
Target entity description: The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.

A. Weierstrass approximation theorem
The Weierstrass approximation theorem is a fundamental result in real analysis stating that any continuous function on a closed interval can be uniformly approximated by polynomials.
B. Banach–Stone theorem
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.
C. 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.
D. 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.
E. 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.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	approximation theorem ⓘ theorem in functional analysis ⓘ
appliesTo	compact Hausdorff spaces ⓘ subalgebras of C(X) ⓘ
assumes	A contains the constant functions ⓘ A is a subalgebra of C(X) ⓘ A separates points of X ⓘ X is a compact Hausdorff space ⓘ
characterizes	density of subalgebras in C(X) ⓘ
complexVersionConclusion	A is uniformly dense in C(X,ℂ) ⓘ
complexVersionCondition	A contains the constant functions ⓘ A is closed under complex conjugation ⓘ A separates points of X ⓘ
concerns	algebras of complex-valued continuous functions ⓘ algebras of real-valued continuous functions ⓘ subalgebras closed under pointwise addition and multiplication ⓘ uniform approximation of continuous functions ⓘ
concludes	A is dense in C(X) with respect to the uniform norm ⓘ every continuous function can be uniformly approximated by elements of A ⓘ
domain	continuous functions on compact spaces ⓘ
field	approximation theory ⓘ functional analysis ⓘ topology ⓘ
generalizes	Weierstrass approximation theorem ⓘ polynomial approximation on compact intervals ⓘ
hasVariant	complex Stone–Weierstrass theorem ⓘ linked to: Stone–Weierstrass theorem real Stone–Weierstrass theorem ⓘ linked to: Stone–Weierstrass theorem
historicalContext	20th-century development in functional analysis ⓘ
implies	density of polynomials in C([a,b]) ⓘ trigonometric polynomial approximation on the circle ⓘ
introducedBy	Marshall Harvey Stone ⓘ linked to: Marshall H. Stone
motivation	extension of polynomial approximation to general compact spaces ⓘ
namedAfter	Karl Weierstrass ⓘ Marshall Harvey Stone ⓘ linked to: Marshall H. Stone
realVersionConclusion	A is uniformly dense in C(X,ℝ) ⓘ
realVersionCondition	A contains the constant functions ⓘ A separates points of X ⓘ
relatedTo	Banach algebras ⓘ linked to: Banach algebra C(X) as a commutative C*-algebra ⓘ Gelfand representation theory ⓘ
topology	uniform norm topology ⓘ
typeOfDensity	uniform density ⓘ
usedIn	C*-algebra theory ⓘ harmonic analysis ⓘ potential theory ⓘ probability theory on compact spaces ⓘ spectral theory ⓘ

How these facts were elicited

Referenced by (5)

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

Weierstrass approximation theorem → generalizationOf → Stone–Weierstrass theorem ⓘ

Weierstrass approximation theorem → relatedTo → Stone–Weierstrass theorem ⓘ

Stone–Weierstrass theorem → hasVariant → real Stone–Weierstrass theorem ⓘ

linked to: Stone–Weierstrass theorem

Stone–Weierstrass theorem → hasVariant → complex Stone–Weierstrass theorem ⓘ

linked to: Stone–Weierstrass theorem

Runge approximation theorem → relatedTo → Stone–Weierstrass theorem ⓘ

All labels observed (3)

How this entity was disambiguated Show

Statements (47)

How these facts were elicited Show

Referenced by (5)

How this entity was disambiguated

How these facts were elicited