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单词 LipschitzCondition
释义

Lipschitz condition


A mapping f:XY between metric spaces is said to satisfy theLipschitz conditionMathworldPlanetmath, or to be Lipschitz continuous or L-Lipschitz if there exists a real constant L suchthat

dY(f(p),f(q))LdX(p,q),for allp,qX.

The least constant L for which the previous inequalityMathworldPlanetmath holds, is called the Lipschitz constant of f.The space of Lipschitz continuous functions is often denoted by Lip(X,Y).

Clearly, every Lipschitz continuous function is continuousMathworldPlanetmath.

Notes.

More generally, one says that a mapping satisfiesa Lipschitz condition of order α>0 if there exists a real constant C such that

dY(f(p),f(q))CdX(p,q)α,for allp,qX.

Functions which satisfy this condition are also called Hölder continuous or α-Hölder. The vector space of such functions is denoted by C0,α(X,Y) and hence Lip=C0,1.

TitleLipschitz condition
Canonical nameLipschitzCondition
Date of creation2013-03-22 11:57:48
Last modified on2013-03-22 11:57:48
Ownerpaolini (1187)
Last modified bypaolini (1187)
Numerical id27
Authorpaolini (1187)
Entry typeDefinition
Classificationmsc 26A16
SynonymLipschitz
SynonymLipschitz continuous
Related topicRademachersTheorem
Related topicNewtonsMethod
Related topicKantorovitchsTheorem
DefinesHolder
DefinesHolder continuous
DefinesLipschitz constant
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