Difference between revisions of "Metric space"
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*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> | *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== | ||
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* 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 | ||
| − | * 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]] | [[Category:Analysis]] | ||
{{stub}} | {{stub}} | ||
Revision as of 13:37, 17 October 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
, with the "usual" meaning of distance- 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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