Difference between revisions of "Metric space"

m
m
Line 3: Line 3:
 
*Symmetry: for all <math>x, y \in S</math>, <math>d(x, y) = d(y, x)</math>
 
*Symmetry: for all <math>x, y \in S</math>, <math>d(x, y) = d(y, x)</math>
 
*Non-negativity: for all <math>x, y \in S</math>, <math>d(x, y) \geq 0</math>
 
*Non-negativity: for all <math>x, y \in S</math>, <math>d(x, y) \geq 0</math>
*Uniqueness (<math>d(x, y) = 0</math> if and only if <math>x = y</math>)
+
*Uniqueness: for all <math>x, y \in S</math>, <math>d(x, y) = 0</math> if and only if <math>x = y</math>
*[[Triangle Inequality]] (<math>d(x, y) + d(y, z) \geq d(x, z)</math> for all points <math>x, y, z \in S</math>).
+
*The [[Triangle Inequality]]:  for all points <math>x, y, z \in S</math>, <math>d(x, y) + d(y, z) \geq d(x, z)</math>  
  
 
Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades).  The above properties follow from our notion of distance.  Non-negativity stems from the idea that  A  cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and  the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.
 
Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades).  The above properties follow from our notion of distance.  Non-negativity stems from the idea that  A  cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and  the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.

Revision as of 08:53, 30 November 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 $S$ which has the following properties:

  • Symmetry: for all $x, y \in S$, $d(x, y) = d(y, x)$
  • Non-negativity: for all $x, y \in S$, $d(x, y) \geq 0$
  • Uniqueness: for all $x, y \in S$, $d(x, y) = 0$ if and only if $x = y$
  • The Triangle Inequality: for all points $x, y, z \in S$, $d(x, y) + d(y, z) \geq d(x, z)$

Intuitively, a metric space is a generalization of the distance between two objects (where "objects" can be anything, including points, functions, graphics, or grades). The above properties follow from our notion of distance. Non-negativity stems from the idea that A cannot be closer to B than B is to itself; Uniqueness results from two objects being identical if and only if they are the same object; and the Triangle Inequality corresponds to the idea that a direct path between points A and B should be at least as short as a roundabout path that visits some point C first.

Popular metrics

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

Invalid username
Login to AoPS