Whitney approximation theorem

E53941

mathematical theorem theorem in differential topology

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.

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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 \(\mathbb{R}^{2n}\)).
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.
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.
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.
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.
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	Weierstrass approximation theorem ⓘ surface form: 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	Whitney approximation theorem self-linksurface differs ⓘ surface form: strong Whitney approximation theorem Whitney approximation theorem self-linksurface differs ⓘ surface form: weak 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	Weierstrass approximation theorem ⓘ surface form: Stone–Weierstrass 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

Referenced by (7)

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

Whitney embedding theorem → relatedTo → Whitney approximation theorem ⓘ

Hassler Whitney → notableFor → Whitney approximation theorem ⓘ

Hassler Whitney → hasTheoremNamedAfter → Whitney approximation theorem ⓘ

Whitney approximation theorem → hasVersion → Whitney approximation theorem self-linksurface differs ⓘ

this entity surface form: weak Whitney approximation theorem

Whitney approximation theorem → hasVersion → Whitney approximation theorem self-linksurface differs ⓘ

this entity surface form: strong Whitney approximation theorem

Hassler → notableFor → Whitney approximation theorem ⓘ

subject surface form: Hassler Whitney

M. Hirsch, Differential Topology → topic → Whitney approximation theorem ⓘ

subject surface form: Differential Topology (book)

All labels observed (3)

How this entity was disambiguated Show

Statements (48)

How these facts were elicited Show

Referenced by (7)

How this entity was disambiguated

How these facts were elicited