Stone–Weierstrass theorem
E480871
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Stone–Weierstrass theorem canonical | 2 |
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]
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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.
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
polynomial approximation on compact intervals ⓘ |
| hasVariant |
complex Stone–Weierstrass theorem
NERFINISHED
ⓘ
real Stone–Weierstrass theorem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| motivation | extension of polynomial approximation to general compact spaces ⓘ |
| namedAfter |
Karl Weierstrass
NERFINISHED
ⓘ
Marshall Harvey Stone NERFINISHED ⓘ |
| realVersionConclusion | A is uniformly dense in C(X,ℝ) ⓘ |
| realVersionCondition |
A contains the constant functions
ⓘ
A separates points of X ⓘ |
| relatedTo |
Banach algebras
NERFINISHED
ⓘ
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
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Instruction
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Input
Subject: Stone–Weierstrass theorem Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.