Difference between revisions of "Metric space"

 
m
Line 1: Line 1:
 
A '''metric space''' is a pair, <math>(S, d)</math> of a [[set]] <math>S</math> and a [[metric]] <math>d: S \times S \to \mathbb{R}_{\geq 0}</math>.  The metric <math>d</math> represents a distance function between pairs of points of <math>S</math>, and we require it to obey symmetry (<math>d(x, y) = d(y, x)</math>), positivity (<math>d(x, y) \geq 0</math> and <math>d(x, y) = 0</math> if and only if <math>x = y</math>) and the [[triangle inequality]] (<math>d(x, y) + d(y, z) \geq d(x, z)</math> for all points <math>x, y, z \in S</math>).
 
A '''metric space''' is a pair, <math>(S, d)</math> of a [[set]] <math>S</math> and a [[metric]] <math>d: S \times S \to \mathbb{R}_{\geq 0}</math>.  The metric <math>d</math> represents a distance function between pairs of points of <math>S</math>, and we require it to obey symmetry (<math>d(x, y) = d(y, x)</math>), positivity (<math>d(x, y) \geq 0</math> and <math>d(x, y) = 0</math> if and only if <math>x = y</math>) and the [[triangle inequality]] (<math>d(x, y) + d(y, z) \geq d(x, z)</math> for all points <math>x, y, z \in S</math>).
 +
 +
==Popular metrics==
 +
 +
* The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance
 +
 +
* The [[discrete metric]] on any set
  
 
{{stub}}
 
{{stub}}

Revision as of 16:56, 23 September 2006

A metric space is a pair, $(S, d)$ of a set $S$ and a metric $d: S \times S \to \mathbb{R}_{\geq 0}$. The metric $d$ represents a distance function between pairs of points of $S$, and we require it to obey symmetry ($d(x, y) = d(y, x)$), positivity ($d(x, y) \geq 0$ and $d(x, y) = 0$ if and only if $x = y$) and the triangle inequality ($d(x, y) + d(y, z) \geq d(x, z)$ for all points $x, y, z \in S$).

Popular metrics

This article is a stub. Help us out by expanding it.

Invalid username
Login to AoPS