# Difference between revisions of "Metric space"

m |
m (→Popular metrics) |
||

(4 intermediate revisions by 4 users not shown) | |||

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 | + | *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]] | + | *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 | + | 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== | ==Popular metrics== | ||

− | * The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance | + | * The [[Euclidean metric]] on <math>\mathbb{R}^n</math>, with the "usual" meaning of distance which is given by <math>d(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2}</math> where <math>x=(x_1,x_2,\dots, x_n)</math> and <math>y=(y_1,y_2,\dots ,y_n)</math>. |

− | * The [[Discrete metric]] on any set | + | * The [[Discrete metric]] on any set, where <math>d(x,y)=1</math> if and only if <math>x\neq y</math> |

+ | * The [[Taxicab metric]] on <math>\mathbb{R}^2</math>, with <math>d(((x_1,y_1),(x_2,y_2))=|x_1-x_2|+|y_1-y_2|</math> | ||

+ | |||

+ | [[Category:Analysis]] | ||

{{stub}} | {{stub}} |

## Latest revision as of 00:19, 22 December 2012

A **metric space** is a pair, of a set and a metric . The metric represents a distance function between pairs of points of which has the following properties:

- Symmetry: for all ,
- Non-negativity: for all ,
- Uniqueness: for all , if and only if
- The Triangle Inequality: for all points ,

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

- The Euclidean metric on , with the "usual" meaning of distance which is given by where and .

- The Discrete metric on any set, where if and only if

- The Taxicab metric on , with

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