Metric

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.