In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

theory of

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

s, a metric map is a function between metric spaces that does not increase any distance. These maps are the

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s in the category of metric spaces, Met. Such functions are always

continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...

s. They are also called

Lipschitz functions In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...

with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps. Specifically, suppose that

X

and

Y

are metric spaces and

f

is a function from

X

Y

. Thus we have a metric map when,

for any In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by ev ...

points

x

and

y

X

d_(f(x),f(y)) \leq d_(x,y) . \!

Here

d_X

and

d_Y

denote the metrics on

X

and

Y

respectively.

Examples

Consider the metric space

,1/2 /math> with the

Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is oc ...

. Then the function

f(x)=x^2

is a metric map, since for

x\ne y

, f(x)-f(y), =, x+y, , x-y, <, x-y,

Category of metric maps

The

function composition In mathematics, the composition operator \circ takes two function (mathematics), functions, f and g, and returns a new function h(x) := (g \circ f) (x) = g(f(x)). Thus, the function is function application, applied after applying to . (g \c ...

of two metric maps is another metric map, and the

identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

\mathrm_M\colon M \rightarrow M

on a metric space

M

is a metric map, which is also the

identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

for function composition. Thus metric spaces together with metric maps form a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

Met. Met is a

subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...

of the category of metric spaces and Lipschitz functions. A map between metric spaces is an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

if and only if it is a

bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...

metric map whose inverse is also a metric map. Thus the

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

s in Met are precisely the isometries.

Multivalued version

A mapping

T\colon X\to \mathcal(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 L d(x,y),

for all

x,y\in X

, where

H

is the

Hausdorff distance In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets o ...

. When

L=1

T

is called ''nonexpansive'', and when

L<1

T

is called a contraction.

References

{{Topology Lipschitz maps Metric geometry Theory of continuous functions

Examples

Category of metric maps

Multivalued version

See also

References