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

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 ⓘ

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)