In themathematical theory ofmetric spaces, ametric map is afunction between metric spaces that does not increase any distance. These maps are themorphisms in thecategory of metric spaces,Met.^[1] Such functions are alwayscontinuous functions.They are also calledLipschitz functions withLipschitz constant 1,nonexpansive maps,nonexpanding maps,weak contractions, orshort maps.

Specifically, suppose that${\displaystyle X}$ and${\displaystyle Y}$ are metric spaces and${\displaystyle f}$ is afunction from${\displaystyle X}$ to${\displaystyle Y}$. Thus we have a metric map when,for any points${\displaystyle x}$ and${\displaystyle y}$ in${\displaystyle X}$,$${\displaystyle d_Y(f(x),f(y))\le d_X(x,y).}$$Here${\displaystyle d_X}$ and${\displaystyle d_Y}$ denote the metrics on${\displaystyle X}$ and${\displaystyle Y}$ respectively.

Consider the metric space${\displaystyle [0,1/2]}$ with theEuclidean metric. Then the function${\displaystyle f(x)=x^2}$ is a metric map, since for${\displaystyle x\ne y}$,.

Thefunction composition of two metric maps is another metric map, and theidentity map${\displaystyle {\mathrm{i}\mathrm{d}}_M:M\to M}$ on a metric space${\displaystyle M}$ is a metric map, which is also theidentity element for function composition. Thus metric spaces together with metric maps form acategoryMet.Met is asubcategory of the category of metric spaces and Lipschitz functions. A map between metric spaces is anisometry if and only if it is abijective metric map whoseinverse is also a metric map. Thus theisomorphisms inMet are precisely the isometries.

One can say that${\displaystyle f}$ isstrictly metric if theinequality is strict for every two different points. Thus acontraction mapping is strictly metric, but not necessarily the other way around. Note that an isometry isnever strictly metric, except in thedegenerate case of theempty space or a single-point space.

A mapping${\displaystyle T:X\to \mathcal{N}(X)}$ from a metric space${\displaystyle X}$ to the family of nonempty subsets of${\displaystyle X}$ is said to be Lipschitz if there exists${\displaystyle L\ge 0}$ such that$${\displaystyle H(Tx,Ty)\le Ld(x,y),}$$for all${\displaystyle x,y\in X}$, where${\displaystyle H}$ is theHausdorff distance. When${\displaystyle L=1}$,${\displaystyle T}$ is callednonexpansive, and when,${\displaystyle T}$ is called acontraction.

• Contraction (operator theory) – Bounded operators with sub-unit norm
• Contraction mapping – Function reducing distance between all points
• Stretch factor – Mathematical parameter of embeddings
• Subcontraction map – Function reducing distance between all pointsPages displaying short descriptions of redirect targets

• Lipschitz maps
• Metric geometry
• Theory of continuous functions

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