Whitney approximation theorem
E53941
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Whitney approximation theorem canonical | 5 |
| strong Whitney approximation theorem | 1 |
| weak Whitney approximation theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T429562 — 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: Whitney approximation theorem Context triple: [Whitney embedding theorem, relatedTo, Whitney approximation theorem]
-
A.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
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B.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
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C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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D.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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E.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
- 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: Whitney approximation theorem Target entity description: The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
A.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
B.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
-
C.
Brouwer fixed-point theorem
The Brouwer fixed-point theorem is a fundamental result in topology stating that any continuous function from a compact convex set (such as a closed disk) to itself has at least one fixed point.
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D.
Janet–Cartan theorem
The Janet–Cartan theorem is a fundamental result in differential geometry stating that any real-analytic Riemannian manifold can be locally isometrically embedded into a Euclidean space of sufficiently high dimension.
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E.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in differential topology ⓘ |
| appearsIn |
textbooks on differential topology
ⓘ
textbooks on smooth manifolds ⓘ |
| asserts |
continuous functions can be approximated arbitrarily well by smooth functions in the compact-open topology
ⓘ
every continuous map between smooth manifolds can be uniformly approximated by smooth maps ⓘ for manifolds without boundary, continuous maps can be approximated by smooth maps that are homotopic to the original map ⓘ |
| assumes |
paracompactness of manifolds in standard formulations
ⓘ
smooth structure on manifolds ⓘ |
| classification |
approximation theorem
ⓘ
linked to:
Weierstrass approximation theorem
|
| codomainCondition | target is a smooth manifold ⓘ |
| concerns |
approximation of continuous maps by smooth maps
ⓘ
continuous functions between smooth manifolds ⓘ smooth manifolds ⓘ |
| domainCondition | source is a smooth manifold ⓘ |
| field |
differential geometry
ⓘ
differential topology ⓘ topology ⓘ |
| generalizes | approximation of continuous functions on subsets of Euclidean space by smooth functions ⓘ |
| hasVersion |
strong Whitney approximation theorem
ⓘ
linked to:
Whitney approximation theorem
weak Whitney approximation theorem ⓘ
linked to:
Whitney approximation theorem
|
| historicalPeriod | 20th century mathematics ⓘ |
| holdsFor | maps between manifolds with boundary under suitable compatibility conditions ⓘ |
| implies |
C^∞(M,N) is dense in C^0(M,N) for smooth manifolds M and N under suitable topologies
ⓘ
smooth maps are dense in the space of continuous maps between smooth manifolds ⓘ |
| mathematicsSubjectClassification |
57Rxx
ⓘ
58Cxx ⓘ |
| namedAfter | Hassler Whitney ⓘ |
| namedEntityType | result in mathematics ⓘ |
| relatedTo |
Stone–Weierstrass theorem
ⓘ
linked to:
Weierstrass approximation theorem
Weierstrass approximation theorem ⓘ Whitney embedding theorem ⓘ |
| standardReference |
J. Lee, Introduction to Smooth Manifolds
ⓘ
J. Munkres, Elementary Differential Topology ⓘ M. Hirsch, Differential Topology ⓘ |
| strongVersionConcerns | approximation relative to a closed subset ⓘ |
| strongVersionStates | a continuous map can be approximated by a smooth map that agrees with it on a closed subset where it is already smooth ⓘ |
| topologyUsed |
C^0 topology on spaces of maps
ⓘ
compact-open topology ⓘ |
| typicalProofUses |
local coordinate charts
ⓘ
partitions of unity ⓘ smoothing by convolution in Euclidean space ⓘ |
| usedIn |
approximation of sections of fiber bundles
ⓘ
construction of smooth structures ⓘ differential topology proofs involving transversality ⓘ homotopy theory of manifolds ⓘ smoothing of continuous maps ⓘ |
| weakVersionStates | every continuous map between smooth manifolds can be uniformly approximated by smooth maps ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Whitney approximation theorem Description of subject: The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Whitney approximation theorem
linked to: Whitney approximation theorem
subject linked to:
Hassler
subject linked to:
M. Hirsch, Differential Topology