Difference between revisions of "Metric space"
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==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" <math>d:X\times X\rightarrow\mathbb{R}</math> | + | * <math>d:X\times X\rightarrow\mathbb{R}</math> or sometimes |
| − | + | * <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> | + | # <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> | ||
| − | + | {{Begin Theorem}} | |
| − | + | Proof that this is a metric | |
| − | {{Todo | + | {{Begin Proof}} |
| + | {{Todo}} | ||
| + | {{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} | ||
| − | + | 0 & x=y\\ | |
| − | + | 1 & \text{otherwise} | |
\end{array}\right.</math> | \end{array}\right.</math> | ||
| + | {{Begin Theorem}} | ||
| + | Proof that this is a metric | ||
| + | {{Begin Proof}} | ||
| + | {{Todo|Really easy though}} | ||
| + | {{End Proof}}{{End Theorem}} | ||
| + | |||
| + | ==See also== | ||
| + | * [[Topological space]] | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Topology|Metric Space}} | {{Definition|Topology|Metric Space}} | ||
Revision as of 00:30, 22 June 2015
Contents
[hide]Definition of a metric space
A metric space is a set X coupled with a "distance function"[1]:
- d:X×X→R or sometimes
- d:X×X→R+[2]
With the properties that for x,y,z∈X:
- d(x,y)≥0
- d(x,y)=0⟺x=y
- d(x,y)=d(y,x) - Symmetry
- 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×X→R) 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)=√n∑i=1((xi−yi)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).
ddiscreet(x,y)={0x=y1otherwise
[Expand]
Proof that this is a metric
