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.

Aliases (2)

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 →
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 → 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	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 (5)

Subject (surface form when different)	Predicate
Whitney approximation theorem ("weak Whitney approximation theorem") → Whitney approximation theorem ("strong Whitney approximation theorem") →	hasVersion
Hassler Whitney →	hasTheoremNamedAfter
Hassler Whitney →	notableFor
Whitney embedding theorem →	relatedTo