Difference between revisions of "Metric space"

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m (Euclidian Metric)
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==Definition of a metric space==
 
==Definition of a metric space==
 
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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" <math>d:X\times X\rightarrow\mathbb{R}</math> with the properties (for <math>x,y,z\in X</math>)
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* <math>d:X\times X\rightarrow\mathbb{R}</math> or sometimes
 
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* <math>d:X\times X\rightarrow\mathbb{R}_+</math><ref name="Analysis">Analysis - Part 1: Elements - Krzysztof Maurin</ref>
 +
With the properties that for <math>x,y,z\in X</math>:
 
# <math>d(x,y)\ge 0</math>
 
# <math>d(x,y)\ge 0</math>
 
# <math>d(x,y)=0\iff x=y</math>
 
# <math>d(x,y)=0\iff x=y</math>
# <math>d(x,y)=d(y,x)</math>
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# <math>d(x,y)=d(y,x)</math> - Symmetry
 
# <math>d(x,z)\le d(x,y)+d(y,z)</math> - the [[Triangle inequality]]
 
# <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>
 
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==
 
==Examples of metrics==
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<math>d_{\text{Euclidian}}(x,y)=\sqrt{\sum^n_{i=1}((x_i-y_i)^2)}</math>
 
<math>d_{\text{Euclidian}}(x,y)=\sqrt{\sum^n_{i=1}((x_i-y_i)^2)}</math>
 
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{{Begin Theorem}}
====Proof it is a metric====
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Proof that this is a metric
{{Todo|Proof this is a metric}}
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{{Begin Proof}}
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{{Todo}}
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{{End Proof}}{{End Theorem}}
  
 
===Discreet Metric===
 
===Discreet Metric===
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<math>d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr}
 
<math>d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr}
       1 & x=y\\
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       0 & x=y\\
       0 & \text{otherwise}
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       1 & \text{otherwise}
 
     \end{array}\right.</math>
 
     \end{array}\right.</math>
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{{Begin Theorem}}
 +
Proof that this is a metric
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{{Begin Proof}}
 +
{{Todo|Really easy though}}
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{{End Proof}}{{End Theorem}}
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 +
==See also==
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* [[Topological space]]
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 +
==References==
 +
<references/>
  
 
{{Definition|Topology|Metric Space}}
 
{{Definition|Topology|Metric Space}}

Revision as of 00:30, 22 June 2015

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]

Proof that this is a metric




TODO:



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].

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

Proof that this is a metric




TODO: Really easy though



See also

References

  1. Introduction to Topology - Bert Mendelson
  2. Analysis - Part 1: Elements - Krzysztof Maurin