Short map
Encyclopedia
In the mathematical
theory of metric space
s, a metric map is a function
between metric spaces that does not increase any distance (such functions are always continuous
).
These maps are the morphism
s 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. Then we have a metric map when, for any points x and y in X,
Here dX and dY denote the metrics on X and Y respectively.
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 isomorphism
s in Met are the isometries.
One can say that ƒ is strictly metric if the inequality is strict for every two different points. Then 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
theory of metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
s, a metric map is a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
between metric spaces that does not increase any distance (such functions are always continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
).
These maps are the morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s in the category of metric spaces
Category of metric spaces
In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...
, Met (Isbell 1964).
They are also called Lipschitz functions
Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: for every pair of points on the graph of this function, the absolute value of the...
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
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from X to Y. Then we have a metric map when, for any points x and y in X,
Here dX and dY denote the metrics on X and Y respectively.
A map ƒ between metric spaces is an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
if and only if 1) it is metric, 2) it is a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
, and 3) its inverse is also metric. The composite of metric maps is also metric. Thus metric spaces and metric maps form a category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
Met
Category of metric spaces
In category-theoretic mathematics, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map...
; Met is a subcategory
Subcategory
In mathematics, 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, a subcategory of C is a category obtained from C by "removing" some of its objects and...
of the category of metric spaces and Lipschitz functions, and the isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
s in Met are the isometries.
One can say that ƒ is strictly metric if the inequality is strict for every two different points. Then a contraction mapping
Contraction mapping
In mathematics, a contraction mapping, or contraction, on a metric space is a function f from M to itself, with the property that there is some nonnegative real number k...
is strictly metric, but not necessarily the other way around. Note that an isometry is never strictly metric, except in the degenerate
Degeneracy (mathematics)
In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class....
case of the empty space
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
or a single-point space.