Lattice (order)

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,

$d_{Y}(f(x),f(y))\leq d_{X}(x,y).\!$

Here d_X and d_Y 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 $T:X\to {\mathcal {N}}(X)$ from a metric space X to the family of nonempty subsets of X is said to be Lipschitz if there exists $L\geq 0$ such that

$H(Tx,Ty)\leq Ld(x,y),$

for all $x,y\in X$, where H is the Hausdorff distance. When $L=1$, T is called nonexpansive and when $L<1$, T is called a contraction.

References

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