Editing Metric space (section)

===Lipschitz maps and contractions===
{{Main|Lipschitz continuity}}

A [[Lipschitz continuity|Lipschitz map]] is one that stretches distances by at most a bounded factor.  Formally, given a real number {{math|''K'' > 0}}, the map <math>f\,\colon M_1\to M_2</math> is {{mvar|K}}-''Lipschitz'' if
<math display="block">d_2(f(x),f(y))\leq K d_1(x,y)\quad\text{for all}\quad x,y\in M_1.</math>
Lipschitz maps are particularly important in metric geometry, since they provide more flexibility than distance-preserving maps, but still make essential use of the metric.{{sfn|Gromov|2007|p=xvii}}  For example, a curve in a metric space is [[arc length|rectifiable]] (has finite length) if and only if it has a Lipschitz reparametrization.

A 1-Lipschitz map is sometimes called a ''nonexpanding'' or ''[[metric map]]''.  Metric maps are commonly taken to be the morphisms of the [[category of metric spaces]].

A {{mvar|K}}-Lipschitz map for {{math|''K'' < 1}} is called a ''[[contraction mapping|contraction]]''.  The [[Banach fixed-point theorem]] states that if {{mvar|M}} is a complete metric space, then every contraction <math>f:M \to M</math> admits a unique [[fixed point (mathematics)|fixed point]]. If the metric space {{mvar|M}} is compact, the result holds for a slightly weaker condition on {{mvar|f}}: a map <math>f:M \to M</math> admits a unique fixed point if
<math display="block"> d(f(x), f(y)) < d(x, y) \quad \mbox{for all} \quad x \ne y \in M_1.</math>