Difference between revisions of "Metric space"

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 [math]X[/math] coupled with a "distance function"^[1]:

• [math]d:X\times X\rightarrow\mathbb{R}[/math] or sometimes
• [math]d:X\times X\rightarrow\mathbb{R}_+[/math]^[2]

With the properties that for [math]x,y,z\in X[/math]:

1. [math]d(x,y)\ge 0[/math]
2. [math]d(x,y)=0\iff x=y[/math]
3. [math]d(x,y)=d(y,x)[/math] - Symmetry
4. [math]d(x,z)\le d(x,y)+d(y,z)[/math] - the Triangle inequality

We will denote a metric space as [math](X,d)[/math] (as [math](X,d:X\times X\rightarrow\mathbb{R})[/math] is too long and Mathematicians are lazy) or simply [math]X[/math] if it is obvious which metric we are talking about on [math]X[/math]

Examples of metrics

Euclidian Metric

The Euclidian metric on [math]\mathbb{R}^n[/math] is defined as follows: For [math]x=(x_1,...,x_n)\in\mathbb{R}^n[/math] and [math]y=(y_1,...,y_n)\in\mathbb{R}^n[/math] we define the Euclidian metric by:

[math]d_{\text{Euclidian}}(x,y)=\sqrt{\sum^n_{i=1}((x_i-y_i)^2)}[/math]

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 [math]X[/math], [math]\mathcal{P}(X)[/math].

It is given by:

• [math]d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr} 0 & x=y\\ 1 & \text{otherwise} \end{array}\right.[/math]

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

See also

• Topological space

References

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