Metric (analysis)

Revision as of 08:26, 8 August 2009 by JBL (talk | contribs) (moved Metric (set theory) to Metric (analysis): Set theory? Gimme a break ... (though if someone thought "topology" was better than "analysis," I wouldn't mind))

A metric $d$ on a set $S$ is a function $d: S \times S \to \mathbb{R}$ which obeys the following three properties:

  • Symmetry: $d(x, y) = d(y, x)$ for all points $x, y \in S$.
  • Positivity: $d(x, y) \geq 0$ for all $x, y \in S$ and $d(x, y) = 0$ if and only if $x = y$.
  • The triangle inequality: $d(x, y) + d(y, z) \geq d(x, z)$ for all $x, y, z \in S$.

Together, the set $S$ and the metric $d$ form a metric space.

Common metrics

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