Short map

From Academic Kids

In mathematics, a short map, or nonexpansive map, is a special kind of continuous function between metric spaces.

Specifically, suppose that X and Y are metric spaces and f is a function from X to Y. Then we have a short map when, for any points x and y in X,

<math> d_{Y}(f(x),f(y)) \leq d_{X}(x,y) . \! <math>

Here dX and dY denote the metrics on X and Y, respectively.

The function f is short if and only if it is 1-Lipschitz continuous. f is an isometry if and only if it is short, it is a bijection, and its inverse is also short. (Because of the inverse function the inequality must become an equality.) The composite of short maps is also short. Thus metric spaces and short maps form a category Met; Met is a subcategory of the category of metric spaces and Lipschitz maps, and the isomorphisms in Met are the isometries.

One can say that f is strictly short if the inequality is always strict. Then a contraction mapping is strictly short, but not necessarily the other way around. Note that an isometry is never strictly short (except in the degenerate case of the empty space).

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