Difference between revisions of "Metric space"

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A [[Normed space|normed space]] is a special case of a metric space, to see the relationships between metric spaces and others see: [[Subtypes of topological spaces]]
 
==Definition of a metric space==
 
==Definition of a metric space==
 
A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction to Topology - Bert Mendelson</ref>:
 
A metric space is a set <math>X</math> coupled with a "distance function"<ref name="Topology">Introduction to Topology - Bert Mendelson</ref>:

Revision as of 22:26, 11 July 2015

A normed space is a special case of a metric space, to see the relationships between metric spaces and others see: Subtypes of topological spaces

Definition of a metric space

A metric space is a set X coupled with a "distance function"[1]:

  • d:X×XR or sometimes
  • d:X×XR+[2]

With the properties that for x,y,zX:

  1. d(x,y)0
  2. d(x,y)=0x=y
  3. d(x,y)=d(y,x) - Symmetry
  4. d(x,z)d(x,y)+d(y,z) - the Triangle inequality

We will denote a metric space as (X,d) (as (X,d:X×XR) is too long and Mathematicians are lazy) or simply X if it is obvious which metric we are talking about on X

Examples of metrics

Euclidian Metric

The Euclidian metric on Rn is defined as follows: For x=(x1,...,xn)Rn and y=(y1,...,yn)Rn we define the Euclidian metric by:

dEuclidian(x,y)=ni=1((xiyi)2)

[Expand]

Proof that this is a metric


Discreet Metric

This is a useless metric, but is a metric and induces the Discreet Topology on X, where the topology is the powerset of X, P(X).

It is given by:

  • ddiscreet(x,y)={0x=y1otherwise

Note: it is sometimes called the trivial metric[3]

[Expand]

Proof that this is a metric


See also

References

  1. Jump up Introduction to Topology - Bert Mendelson
  2. Jump up Analysis - Part 1: Elements - Krzysztof Maurin
  3. Jump up Functional Analysis - George Bachman and Lawrence Narici