# Difference between revisions of "Metric"

distance on a set $X$

A function $\rho$ with non-negative real values, defined on the Cartesian product $X\times X$ and satisfying for any $x, y\in X$ the conditions:

1) $\rho(x,y)=0$ if and only if $x = y$ (the identity axiom);

2) $\rho(x,y) + \rho(y,z) \geq \rho(x,z)$ (the triangle axiom);

3) $\rho(x,y) = \rho(y,x)$ (the symmetry axiom).

A set $X$ on which it is possible to introduce a metric is called metrizable (cf. Metrizable space). A set $X$ provided with a metric is called a metric space.

## Contents

### Examples.

1) On any set there is the discrete metric \begin{equation} \rho(x,y) = 0 \text{ if } x=y \quad \text{and} \quad \rho(x,y) = 1 \text{ if } x\ne y. \end{equation}

2) In the space $\mathbb R^n$ various metrics are possible, among them are: \begin{equation} \rho(x,y) = \sqrt{\sum(x_i-y_i)^2}; \end{equation} \begin{equation} \rho(x,y)=\sup\limits_i|x_i-y_i|; \end{equation} \begin{equation} \rho(x,y)=\sum|x_i-y_i|; \end{equation}

here $\{x_i\}, \{y_i\} \in \mathbb{R}^n$.

3) In a Riemannian space a metric is defined by a metric tensor, or a quadratic differential form (in some sense, this is an analogue of the first metric of example 2)). For a generalization of metrics of this type see Finsler space.

4) In function spaces on a (countably) compact space $X$ there are also various metrics; for example, the uniform metric \begin{equation} \rho(f,g)=\sup\limits_{x\in X}|f(x)-g(x)| \end{equation}

(an analogue of the second metric of example 2)), and the integral metric \begin{equation} \rho(f,g)=\int\limits_X|f-g|\, dx. \end{equation}

5) In normed spaces over $\mathbb R$ a metric is defined by the norm $\|\cdot\|$: \begin{equation} \rho(x,y) = \|x-y\|. \end{equation}

6) In the space of closed subsets of a metric space there is the Hausdorff metric.

If, instead of 1), one requires only:

1') $\rho(x,y)=0$ if $x=y$ (so that from $\rho(x,y)=0$ it does not always follows that $x=y$), the function $\rho$ is called a pseudo-metric , , or finite écart .

A metric (and even a pseudo-metric) makes the definition of a number of additional structures on the set $X$ possible. First of all a topology (see Topological space), and in addition a uniformity (see Uniform space) or a proximity (see Proximity space) structure. The term metric is also used to denote more general notions which do not have all the properties 1)–3); such are, for example, an indefinite metric, a symmetry on a set, etc.