Difference between revisions of "Metric space"

Revision as of 20:33, 23 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], Note that here I prefer the notation [math]d:X\times X\rightarrow\mathbb{R}_{\ge 0}[/math]

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

1. [math]d(x,y)\ge 0[/math] (This is implicit with the [ilmath]d:X\times X\rightarrow\mathbb{R}_{\ge 0}[/ilmath] definition)
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}_{\ge 0})[/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]

Discrete Metric

Let [ilmath]X[/ilmath] be a set. The discrete^[3] metric, or trivial metric^[4] is the metric defined as follows:

• [math]d:X\times X\rightarrow \mathbb{R}_{\ge 0} [/math] with [math]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\1 & \text{otherwise}\end{array}\right.[/math]

However any strictly positive value will do for the [ilmath]x\ne y[/ilmath] case. For example we could define [ilmath]d[/ilmath] as:

• [math]d:(x,y)\mapsto\left\{\begin{array}{lr}0 & \text{if }x=y \\v & \text{otherwise}\end{array}\right.[/math]
  • Where [ilmath]v[/ilmath] is some arbitrary member of [ilmath]\mathbb{R}_{> 0} [/ilmath]^{[Note 1]} - traditionally (as mentioned) [ilmath]v=1[/ilmath] is used.
    

Note: however in proofs we shall always use the case [ilmath]v=1[/ilmath] for simplicity

Notes

See also

• Topology induced by a metric
• Connected space
• Topological space

Notes

1. ↑ Note the strictly greater than 0 requirement for [ilmath]v[/ilmath]

References

1. ↑ Introduction to Topology - Bert Mendelson
2. ↑ Analysis - Part 1: Elements - Krzysztof Maurin
3. ↑ Introduction to Topology - Theodore W. Gamelin & Robert Everist Greene
4. ↑ Functional Analysis - George Bachman and Lawrence Narici