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...unction]] between metric spaces that does not increase any distance. These maps are the [[morphism]]s in the [[category of metric spaces]], '''Met'''.{{r|i ...ive maps''', '''nonexpanding maps''', '''weak contractions''', or '''short maps'''. ...

|name = Rudolph Lipschitz |caption = Rudolf Lipschitz ...

...on|Category whose objects are metric spaces and whose morphisms are metric maps}} ...a category because the [[Function composition|composition]] of two metric maps is again a metric map. It was first considered by {{harvtxt|Isbell|1964}}. ...

[[File:Lipschitz Visualisierung.gif|thumb|right|For a Lipschitz continuous function, there exists a double cone (white) whose origin can be ...ction that is defined on an interval and has a bounded first derivative is Lipschitz continuous.<ref>{{cite book |url=https://books.google.com/books?id=gBPI_oYZ ...

is a [[Lipschitz continuity|Lipschitz-continuous]] map, then there is a Lipschitz-continuous map that extends {{mvar|f}} and has the same Lipschitz constant as {{mvar|f}}. ...

:* a sequence of <math>(1+\epsilon_i)</math>-bi-Lipschitz [[diffeomorphism]]s <math>\phi_i: M_{i, [\epsilon_i, \infty)} \rightarrow M :where the domains and ranges of the maps are the <math>\epsilon_i</math>-thick parts of either the <math>M_i</math>' ...

...| title = A proof of the Hilbert-Smith conjecture for actions by Lipschitz maps | periodical = [[Mathematische Annalen]] | volume = 308 | issue = 2 |date=J ...

...countable]] collection <math>\{f_i\}</math> of continuously differentiable maps ...that <math>E</math> is a countable union of sets which are the image of a Lipschitz map from some bounded subset of <math>\mathbb{R}^m</math>.<ref>{{harvtxt|Fe ...

...[topological vector space]]s (TVSs) <math>X</math> and <math>Y</math> that maps [[Bounded set (topological vector space)|bounded]] subsets of <math>X</math Every bounded operator is [[Lipschitz continuity|Lipschitz continuous]] at <math>0.</math> ...

...y]], the modulus ω(''t'') := ''kt''(|log ''t''|+1) describes the '''almost Lipschitz''' class, and so on. In general, the role of ω is to fix some explicit func ...taining ''X''. Also, it can be characterized as the uniform closure of the Lipschitz functions on ''X''. ...

...tant]]''' of ''f''. Contractive maps are sometimes called '''Lipschitzian maps'''. If the above condition is instead satisfied for More generally, the idea of a contractive mapping can be defined for maps between metric spaces. Thus, if (''M'',&thinsp;''d'') and (''N'',&thinsp;'' ...

...see proof below). If {{math|1=''α'' = 1}}, then the function satisfies a [[Lipschitz condition]]. For any {{math|''α'' > 0}}, the condition implies the function ...ndent | em = 1.5 | text = '''[[Continuously differentiable]]''' &sub; '''[[Lipschitz continuous]]''' &sub; '''<math>\alpha</math>-Hölder continuous''' &sub; '' ...

...tinuous maps or in the category of uniform spaces and uniformly continuous maps. These facts are examples of a much broader phenomenon, in which discrete s ...y function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric ...

...sub>0</sub> and ''y''₀ and satisfies the [[Lipschitz continuity|Lipschitz condition]] on the variable ''y''. ...nt [[integral equation]]. The integral can be considered an operator which maps one function into another, such that the solution is a [[Fixed point (mathe ...

...ution. It is also known as '''Picard's existence theorem''', the '''Cauchy–Lipschitz theorem''', or the '''existence and uniqueness theorem'''. The theorem is named after [[Émile Picard]], [[Ernst Lindelöf]], [[Rudolf Lipschitz]] and [[Augustin-Louis Cauchy]]. ...

{{Short description|Continuous maps on a closed subset of a normal space can be extended}} ...ipschitz constant <math>K,</math> then <math>f</math> can be extended to a Lipschitz continuous function <math>F : X \to \R</math> with same constant <math>K.</ ...

...d uniqueness of [[fixed point (mathematics)|fixed points]] of certain self-maps of metric spaces and provides a constructive method to find those fixed poi ...]'' for <math>T</math>, and the smallest one is sometimes called "the best Lipschitz constant" of <math>T</math>. ...

...le, [[Analytic function|real or complex analytic]], [[Lipschitz continuity|Lipschitz]], [[Hölder continuity|Hölder]], [[Quasiconformal mapping|quasi-conformal]] ...e whose fibers are vector spaces and whose transition functions are linear maps. ...

...}} itself are vector spaces over {{var|F}}, the set of [[Linear map|linear maps]] {{var|X}} → {{var|V}} form a vector space over {{var|F}} with pointwise o ...one can form a representation of {{var|G}} over the vector space of linear maps Hom({{var|V}},{{var|W}}) called the [[Hom representation]].<ref>{{Cite book ...

*[[co-Lipschitz map]] [[Category:Maps of manifolds]] ...