Difference between revisions of "Metric space"
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==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)=\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, where <math>d(x,y)=1</math> if and only if <math>x\neq y</math> | * The [[Discrete metric]] on any set, where <math>d(x,y)=1</math> if and only if <math>x\neq y</math> | ||
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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