In the
mathematical theory of
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s, a metric map is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between metric spaces that does not increase any distance (such functions are always
continuous).
These maps are the
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s in the
category of metric spaces In category theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of tw ...
, Met (Isbell 1964).
They are also called
Lipschitz functions
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exi ...
with
Lipschitz constant
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exi ...
1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps.
Specifically, suppose that ''X'' and ''Y'' are metric spaces and ƒ is a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
from ''X'' to ''Y''. Thus we have a metric map when,
for any points ''x'' and ''y'' in ''X'',
:
Here ''d''
''X'' and ''d''
''Y'' denote the metrics on ''X'' and ''Y'' respectively.
Examples
Let us consider the metric space