Metric map

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. Then 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.

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.

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.

References

• Isbell, J. R. (1964). "Six theorems about injective metric spaces". Comment. Math. Helv. 39: 65–76. doi:10.1007/BF02566944. http://www.digizeitschriften.de/resolveppn/GDZPPN002058340. 

This geometry-related article is a stub. You can help Wikipedia by expanding it.v · Categories: Metric geometry Lipschitz maps Geometry stubs Wikimedia Foundation. 2010. Игры ⚽ Нужен реферат? Tunnels in Iceland Tailings Look at other dictionaries: Metric space — In mathematics, a metric space is a set where a notion of distance (called a metric) 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… …   Wikipedia Metric space aimed at its subspace — In mathematics, a metric space aimed at its subspace is a categorical construction that has a direct geometric meaning. It is also a useful step toward the construction of the metric envelope, or tight span, which are basic (injective) objects of …   Wikipedia Metric — Metric(s) may refer to: the metric system of measurement International System of Units, or Système International (SI), the modern form of the metric system Metric ton, a measurement of mass equal to 1,000 kg an analytical measurement… …   Wikipedia Map projection — A medieval depiction of the Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy s Geography and using his second map projection A map projection is any method of representing the surface of a sphere or other …   Wikipedia Metric tensor — In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold (such as a surface in space) which takes as input a pair of tangent vectors v and w and produces a real number (scalar) g(v,w) in a… …   Wikipedia Metric (vector bundle) — In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle. Specifically, if M is a topological manifold and E → M a vector bundle on M, then a metric (sometimes called a bundle metric, or fibre metric)… …   Wikipedia Metric differential — In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can… …   Wikipedia Map (disambiguation) — Contents 1 Mathematics and Programming 2 Science 3 Television, film, and music …   Wikipedia metric function — noun a function of a topological space that gives, for any two points in the space, a value equal to the distance between them • Syn: ↑metric • Topics: ↑mathematics, ↑math, ↑maths, ↑physics, ↑natural philosophy …   Useful english dictionary Map of lattices — The concept of a lattice arises in order theory, a branch of mathematics. The Hasse diagram below depicts the inclusion relationships among some important subclasses of lattices. Proofs of the relationships in the map 1. A boolean algebra is a… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”