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In the [[mathematics|mathematical]] theory of [[metric space]]s, a '''metric map''' is a [[Function (mathematics)|function]] between metric spaces that does not increase any distance (such functions are always  [[continuous function|continuous]]).
These maps are the [[morphism]]s in the [[category of metric spaces]], '''Met''' (Isbell 1964).
They are also called [[Lipschitz continuity|Lipschitz functions]] with [[Lipschitz constant]] 1, '''nonexpansive maps''', '''nonexpanding maps''', '''weak contractions''', or '''short maps'''.


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Specifically, suppose that ''X'' and ''Y'' are metric spaces and ƒ is a [[function (mathematics)|function]] from ''X'' to ''Y''. Thus we have a metric map when, [[for any]] points ''x'' and ''y'' in ''X'',
 
: <math> d_{Y}(f(x),f(y)) \leq d_{X}(x,y) . \! </math>
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Here ''d''<sub>''X''</sub> and ''d''<sub>''Y''</sub> denote the metrics on ''X'' and ''Y'' respectively.
 
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==Category of metric maps==
A map ƒ between metric spaces is an [[isometry]] if and only if 1) it is metric, 2) it is a [[bijection]], and 3) its [[inverse (functions)|inverse]] is also metric. The [[composite function|composite]] of metric maps is also metric. Thus metric spaces and metric maps form a [[category theory|category]] '''[[Category of metric spaces|Met]]'''; '''Met''' is a [[subcategory]] of the category of metric spaces and Lipschitz functions, and the [[isomorphism]]s in '''Met''' are the isometries.


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==Strictly metric maps==
 
One can say that ƒ is '''strictly metric''' if the [[inequality (mathematics)|inequality]] is strict for every two different points. Thus a [[contraction mapping]] is strictly metric, but not necessarily the other way around. Note that an isometry is ''never'' strictly metric, except in the [[degeneracy (mathematics)|degenerate]] case of the [[empty set|empty space]] or a single-point space.
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== Multivalued version ==
A mapping <math>T:X\to \mathcal{N}(X)</math> from a metric space ''X'' to the family of nonempty subsets of ''X'' is said to be Lipschitz if there exists <math>L\geq 0</math> such that
:<math>H(Tx,Ty)\leq L d(x,y),</math>
for all <math>x,y\in X</math>, where ''H'' is the [[Hausdorff distance]]. When <math>L=1</math>, ''T'' is called [[nonexpansive]] and when <math>L<1</math>, ''T'' is called a [[contraction]].


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== References ==
 
*{{cite journal
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| author = Isbell, J. R. | authorlink = John R. Isbell
 
| title = Six theorems about injective metric spaces
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| journal = Comment. Math. Helv.
 
| volume = 39
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| year = 1964
 
| pages = 65–76
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| url = http://www.digizeitschriften.de/resolveppn/GDZPPN002058340 | doi = 10.1007/BF02566944
 
}}
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[[Category:Metric geometry]]
 
[[Category:Lipschitz maps]]
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Revision as of 17:17, 31 December 2013

In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous). These maps are the morphisms in the category of metric spaces, Met (Isbell 1964). They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps.

Specifically, suppose that X and Y are metric spaces and ƒ is a function from X to Y. Thus we have a metric map when, for any points x and y in X,

dY(f(x),f(y))dX(x,y).

Here dX and dY denote the metrics on X and Y respectively.

Category of metric maps

A map ƒ between metric spaces is an isometry if and only if 1) it is metric, 2) it is a bijection, and 3) its inverse is also metric. The composite of metric maps is also metric. Thus metric spaces and metric maps form a category Met; Met is a subcategory of the category of metric spaces and Lipschitz functions, and the isomorphisms in Met are the isometries.

Strictly metric maps

One can say that ƒ is strictly metric if the inequality is strict for every two different points. Thus a contraction mapping is strictly metric, but not necessarily the other way around. Note that an isometry is never strictly metric, except in the degenerate case of the empty space or a single-point space.

Multivalued version

A mapping T:X𝒩(X) from a metric space X to the family of nonempty subsets of X is said to be Lipschitz if there exists L0 such that

H(Tx,Ty)Ld(x,y),

for all x,yX, where H is the Hausdorff distance. When L=1, T is called nonexpansive and when L<1, T is called a contraction.

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