Difference between revisions of "Metric space"
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| − | 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> | + | 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> which has the following properties: |
| + | *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> | ||
| + | *Uniqueness: for all <math>x, y \in S</math>, <math>d(x, y) = 0</math> if and only if <math>x = y</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> | ||
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| + | 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. | ||
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| + | ==Popular metrics== | ||
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| + | * 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>. | ||
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| + | * The [[Discrete metric]] on any set, where <math>d(x,y)=1</math> if and only if <math>x\neq y</math> | ||
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| + | * 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> | ||
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| + | [[Category:Analysis]] | ||
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Latest revision as of 01:19, 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
, 
, 
if and only if 
, 
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 
.
, with the "usual" meaning of distance which is given by 
where 
and 
.- The Discrete metric on any set, where
if and only if 
if and only if 
- The Taxicab metric on
, with 
, with 
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