Metric map

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

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

Examples

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

Category of metric maps

The function composition of two metric maps is another metric map, and the identity map ${\displaystyle \mathrm {id} _{M}\colon M\rightarrow M}$ on a metric space ${\displaystyle M}$ is a metric map, which is also the identity element for function composition. Thus metric spaces together with metric maps form a category Met. Met is a subcategory of the category of metric spaces and Lipschitz functions. A map between metric spaces is an isometry if and only if it is a bijective metric map whose inverse is also a metric map. Thus the isomorphisms in Met are precisely the isometries.

Strictly metric maps

One can say that ${\displaystyle f}$ 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 ${\displaystyle T\colon X\to {\mathcal {N}}(X)}$ from a metric space ${\displaystyle X}$ to the family of nonempty subsets of ${\displaystyle X}$ is said to be Lipschitz if there exists ${\displaystyle L\geq 0}$ such that $${\displaystyle H(Tx,Ty)\leq Ld(x,y),}$$ for all ${\displaystyle x,y\in X}$, where ${\displaystyle H}$ is the Hausdorff distance. When ${\displaystyle L=1}$, ${\displaystyle T}$ is called nonexpansive, and when ${\displaystyle L<1}$, ${\displaystyle T}$ is called a contraction.

See also

• 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

References

1. Isbell, J. R. (1964). "Six theorems about injective metric spaces". Comment. Math. Helv. 39: 65–76. doi:10.1007/BF02566944.