Difference between revisions of "Metric space"
m |
m (category) |
||
| Line 14: | Line 14: | ||
* The [[Discrete metric]] on any set | * The [[Discrete metric]] on any set | ||
| + | [[Category:Analysis]] | ||
{{stub}} | {{stub}} | ||
Revision as of 16:47, 28 March 2009
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
This article is a stub. Help us out by expanding it.
