# Metric

A nonnegative, real-valued function, $LaTeX: d$, on pairs from a set $LaTeX: S$ such that for each $LaTeX: x,$ $LaTeX: y,$ and $LaTeX: z$ in $LaTeX: S:$
1. $LaTeX: d(x, y) \ge 0;$
2. $LaTeX: d(x, y) = 0 \Rightarrow x=y;$
3. $LaTeX: d(x, y) + d(y, z) \ge d(x, z).$
The function is also called a distance; the pair $LaTeX: (S, d)$ is a metric space. Condition (3) is called the Triangle inequality. A space is metrizable if a metric can be defined. A metric is induced by a norm if one exists: $LaTeX: d(x,y) = \| x - y \|.$ See Hausdorff metric.