This site is being phased out.
Metric
From Mathematics Is A Science
Revision as of 05:14, 18 February 2011 by imported>WikiSysop
Given a set $X$, any function $d: X \times X \longrightarrow {\bf R}$ is called a metric (or a "distance function") if, for every $x,y,z \in X$,
- $d(x,y) \geq 0$, with equality if and only if $x=y$;
- $d(x,y) = d(y,x)$;
- $d(x,z) \leq d(x,y) + d(y,z)$.
(The last condition is called the triangle inequality.) In this case $(X,d)$ is called a metric space.
The Euclidean space ${\bf R}^n$ has a metric defined by $d(x,y) = ||x-y||$. Similarly, a metric is defined for all normed spaces.