Difference between revisions of "Metric space"

Revision as of 02:59, 19 May 2015 (view source) Alec (Talk | contribs) m (→‎Euclidian Metric) ← Older edit		Revision as of 00:30, 22 June 2015 (view source) Alec (Talk | contribs) m Newer edit →
Line 1:		Line 1:
	==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> with the properties (for <math>x,y,z\in X</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==
Line 18:		Line 18:
	<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 it is a metric====	+	Proof that this is a metric
−	{{Todo|Proof this is a metric}}	+	{{Begin Proof}}
		+	{{Todo}}
		+	{{End Proof}}{{End Theorem}}
	===Discreet Metric===		===Discreet Metric===
Line 26:		Line 28:
	<math>d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr}		<math>d_{\text{discreet}}(x,y)=\left\{\begin{array}{lr}
−	1 & x=y\\	+	0 & x=y\\
−	0 & \text{otherwise}	+	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

Definition of a metric space

A metric space is a set

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

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

With the properties that for

$$x,y,z\in X$$ :

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

We will denote a metric space as

$$(X,d)$$ (as $$(X,d:X\times X\rightarrow\mathbb{R})$$ 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

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

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

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

$$X$$ , $$\mathcal{P}(X)$$ .

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

See also

• Topological space

References

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