Metric space: Difference between revisions

From Citizendium
Jump to navigation Jump to search
imported>Hendra I. Nurdin
m (wording)
mNo edit summary
 
(12 intermediate revisions by 5 users not shown)
Line 1: Line 1:
In [[mathematics]], a '''metric space''' is, roughly speaking, an abstract mathematical structure that generalizes the notion of a Euclidean space <math>\mathbb{R}^n</math> which has been equipped with the Euclidean distance, to more general classes of sets such as a set of functions. A metric space consists of two components, a set and a metric on that set. On a metric space, the metric replaces the Euclidean distance as a notion of "distance" between any pair of elements in its  associated set (for example, as an abstract distance between two functions in a set of functions) and induces a [[topological space|topology]] on the set called the <i>metric topology</i>.
{{subpages}}


== Metric on a set==
In [[mathematics]], a '''metric space''' is, roughly speaking, an abstract mathematical structure that generalizes the notion of a Euclidean space <math>\mathbb{R}^n</math> which has been equipped with the Euclidean distance, to more general classes of sets such as a set of functions. The notion of a metric space consists of two components, a set and a metric in that set. In a metric space, the metric replaces the Euclidean distance as a notion of "distance" between any pair of elements in its  associated set (for example, as an abstract distance between two functions in a set of functions) and induces a [[topological space|topology]] in the set called the <i>metric topology</i>.
Let <math>X</math> be an arbitrary set. A '''metric''' <math>d</math> on <math>X</math> is a function <math>d: X \times X \rightarrow \mathbb{R}</math> with the following properties:


#<math>d(x_1,x_2) \geq 0 \quad \forall x_1,x_2 \in X</math> (non-negativity)
The theory of metric spaces includes the following topics: [[isometric embeddings]] and [[universal metric spaces]] (in the sense of isometric embeddings); [[metric maps]] (which do not increase distances); the [[category (mathematics)|category]] of metric spaces and metric maps, and its subcategories; injective metric spaces and related notions; special classes of metric spaces like [[strong convexity|strongly convex]] spaces; metric generalizations of the notions of [[differential geometry]]; metric properties of the metric spaces which appear in other branches of mathematics (e.g. [[Banach space]]s, in particular [[Hilbert space]]s).
#<math>d(x_2,x_1)=d(x_1,x_2) \quad \forall x_1,x_2 \in X</math> (symmetry)
#<math>d(x_1,x_2)\leq d(x_1,x_3)+d(x_3,x_2) \quad \forall x_1,x_2,x_3 \in X</math>(triangular inequality)
#<math>d(x_1,x_2)=0</math> if and only if <math>x_1=x_2</math>


== Formal definition of metric space ==
The topic of metric spaces can be attractive both to children and to research mathematicians. Certain deeper directions in the theory of metric spaces are closely related to the [[approximation theory]].
A '''metric space''' is an ordered pair <math>(X,d)</math> where <math>X</math> is a set and <math>d</math> is a metric on <math>X</math>.


For shorthand, a metric space <math>(X,d)</math> is usually written simply as <math>X</math> once the metric <math>d</math> has been defined or is understood.
Every [[simple graph|simple]] [[graph]] can be viewed as a metric space (in more than one way). Thus formally the theory of simple graphs can be considered as a special chapter of the theory of metric spaces. Indeed, as a rule, the notions of the theory of simple graphs can be rephrased in the language of metric spaces (possibly in more than one way).
 
== Metric in a set==
Let <math>X\,</math> be an arbitrary set. A '''metric''' <math>d\,</math> on <math>X\,</math> is a function <math>d: X \times X \rightarrow \mathbb{R}</math> &nbsp; with the following properties:
 
#<math>d(x_2,x_1)=d(x_1,x_2) \quad \forall x_1,x_2 \in X</math> &nbsp; (symmetry)
#<math>d(x_1,x_2)\leq d(x_1,x_3)+d(x_3,x_2) \quad \forall x_1,x_2,x_3 \in X</math> &nbsp; (triangular inequality)
#<math>d(x_1,x_2)=0\ \Leftrightarrow \ x_1=x_2\,</math>
 
It follows from the above three axioms of a metric (also called '''distance function''') that:
 
::<math>d(x_1,x_2) \geq 0 \quad \forall x_1,x_2 \in X</math> &nbsp; (non-negativity)
 
== Definition of metric space ==
A '''metric space''' is an ordered pair <math>(X,d)\,</math> where <math>X\,</math> is a set and <math>d\,</math> is a metric on <math>X\,</math>.
 
For shorthand, a metric space <math>(X,d)\,</math> is usually written simply as <math>X\,</math> once the metric <math>d\,</math> has been defined or is understood.


== Metric topology ==
== Metric topology ==
A metric on a set <math>X</math> induces a particular [[topological space|topology]] on <math>X</math> called the ''metric topology''. For any <math>x \in X </math>, let the ''open ball'' <math>B_r(x)</math> of radius <math>r>0</math> around the point <math>x</math> be defined as <math>B_r(x)=\{y \in X \mid d(y,x)<r\}</math>. Define the collection <math>O</math> of subsets of <math>X</math> (meaning that <math>A \in O \Rightarrow A \subset X </math>) consisting of the empty set <math>\emptyset</math> and all sets of the form:
A metric on a set <math>X\,</math> induces a particular [[topological space|topology]] on <math>X\,</math> called the ''metric topology''. For any <math>x \in X </math>, let the ''open ball'' <math>B_r(x)\,</math> of radius <math>r>0\,</math> around the point <math>x\,</math> be defined as <math>B_r(x)=\{y \in X \mid d(y,x)<r\}</math>. Define the collection <math>\mathcal{O}</math> of subsets of <math>X\,</math> (meaning that <math>A \in \mathcal{O} \Rightarrow A \subset X </math>) consisting of the empty set <math>\emptyset</math> and all sets of the form:


<center><math>\cup_{\gamma \in \Gamma}B_{r_{\gamma}}(x_{\gamma}),</math></center>   
<center><math>\cup_{\gamma \in \Gamma}B_{r_{\gamma}}(x_{\gamma}),</math></center>   


where <math>\Gamma</math> is an arbitrary index set (can be uncountable) and <math>r_{\gamma}>0</math> and <math>x_{\gamma} \in X</math> for all <math>\gamma \in \Gamma</math>. Then the set <math>O</math> satisfies all the requirements to be a topology on <math>X</math> and is said to be the ''topology induced by the metric'' <math>d</math>. Any topology induced by a metric is said to be a metric topology.  
where <math>\Gamma\,</math> is an arbitrary index set (can be uncountable) and <math>r_{\gamma}>0\,</math> and <math>x_{\gamma} \in X</math> for all <math>\gamma \in \Gamma</math>. Then the set <math>\mathcal{O}</math> satisfies all the requirements to be a topology on <math>X\,</math> and is said to be the ''topology induced by the metric'' <math>d\,</math>. Any topology induced by a metric is said to be a metric topology.


== Examples ==
== Examples ==
#The "canonical" example of a metric space, and indeed what motivated the general definition of such a space, is the Euclidean space <math>\mathbb{R}^n</math> endowed with the Euclidean distance <math>d_E</math> defined by <math>d_E(x,y)=\sqrt{\sum_{k=1}^{n}|x_k-y_k|^2} \quad \forall x,y \in \mathbb{R}^n </math>.  
#The "canonical" example of a metric space, and indeed what motivated the general definition of such a space, is the Euclidean space <math>\mathbb{R}^n</math> endowed with the Euclidean distance <math>d_E\,</math> defined by <math>d_E(x,y)=\sqrt{\sum_{k=1}^{n}|x_k-y_k|^2}</math> for all <math> x,y \in \mathbb{R}^n </math>.  
#Consider the set <math>C[a,b]</math> of all real valued continuous functions on the interval <math>[a,b]\subset \mathbb{R}</math> with <math>a<b</math>. Define the function <math>d:C[a,b] \times C[a,b] \rightarrow \mathbb{R}</math> by <math>d(f,g)=\max_{x \in [a,b]}|f(x)-g(x)| \quad \forall f,g \in C[a,b]</math>. Then <math>d</math> is a metric on <math>C[a,b]</math> and induces a topology on <math>C[a,b]</math> often known as the ''norm topology'' or ''uniform topology''.
#Consider the set <math>C[a,b]\,</math> of all real valued continuous functions on the interval <math>[a,b]\subset \mathbb{R}</math> with <math>a<b\,</math>. Define the function <math>d:C[a,b] \times C[a,b] \rightarrow \mathbb{R}</math> by <math>d(f,g)=\max_{x \in [a,b]}|f(x)-g(x)| </math> for all <math>f,g \in C[a,b]</math>. This function <math>d\,</math> is a metric on <math>C[a,b]\,</math> and induces a topology on <math>C[a,b]\,</math> often known as the ''norm topology'' or ''uniform topology''.
#Let <math>X\,</math> be any nonempty set. The ''[[discrete metric]]'' on <math>X\,</math> is defined as <math>d(x,y)=1\,</math> if <math>x\neq y</math> and <math>d(x,y)=0\,</math> otherwise. In this case the induced topology is the ''discrete topology'', in which every set is open.
 
==Mappings==
A mapping ''f'' from a metric space (''X'',''d'') to another (''Y'',''e'') is an '''isometry''' if it is distance-preserving: that is
 
:<math>e(f(x_1),f(x_2)) = d(x_1,x_2) . \, </math>
 
A mapping ''f'' from a metric space (''X'',''d'') to another (''Y'',''e'') is ''continuous at'' ''x'' in ''X'' if for all real ε &gt; 0 there exists δ &gt; 0 such that
 
:<math>d(x',x) < \delta \Rightarrow e(f(x'),f(x) < \varepsilon \,</math>
 
and ''continuous'' if it is continuous at every point of ''X''.
 
If we let <math>B_d(x,r)</math> denote the [[open ball]] of radius ''r'' round ''x'' in ''X'', and similarly <math>B_e(y,r)</math> denote the [[open ball]] of radius ''r'' round ''y'' in ''Y'', we can express these conditions in terms of the pull-back <math>f^{\dashv}</math>
 
:<math>f^{\dashv}[B_e(f(x),\varepsilon)] \supseteq B_d(x,\delta) . \, </math>


== See also ==
== See also ==
Line 34: Line 61:


== References ==
== References ==
1. K. Yosida, <i>Functional Analysis</i> (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980
1. K. Yosida, <i>Functional Analysis</i> (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980[[Category:Suggestion Bot Tag]]
 
[[Category:Mathematics Workgroup]]
[[Category:CZ Live]]

Latest revision as of 11:01, 18 September 2024

This article is a stub and thus not approved.
 Main Article
 Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
https://en.citizendium.org/wiki/index.php?title=Metric_space&printable=yes
This editable Main Article is under development and subject to a disclaimer.

In mathematics, a metric space is, roughly speaking, an abstract mathematical structure that generalizes the notion of a Euclidean space which has been equipped with the Euclidean distance, to more general classes of sets such as a set of functions. The notion of a metric space consists of two components, a set and a metric in that set. In a metric space, the metric replaces the Euclidean distance as a notion of "distance" between any pair of elements in its associated set (for example, as an abstract distance between two functions in a set of functions) and induces a topology in the set called the metric topology.

The theory of metric spaces includes the following topics: isometric embeddings and universal metric spaces (in the sense of isometric embeddings); metric maps (which do not increase distances); the category of metric spaces and metric maps, and its subcategories; injective metric spaces and related notions; special classes of metric spaces like strongly convex spaces; metric generalizations of the notions of differential geometry; metric properties of the metric spaces which appear in other branches of mathematics (e.g. Banach spaces, in particular Hilbert spaces).

The topic of metric spaces can be attractive both to children and to research mathematicians. Certain deeper directions in the theory of metric spaces are closely related to the approximation theory.

Every simple graph can be viewed as a metric space (in more than one way). Thus formally the theory of simple graphs can be considered as a special chapter of the theory of metric spaces. Indeed, as a rule, the notions of the theory of simple graphs can be rephrased in the language of metric spaces (possibly in more than one way).

Metric in a set

Let be an arbitrary set. A metric on is a function   with the following properties:

  1.   (symmetry)
  2.   (triangular inequality)

It follows from the above three axioms of a metric (also called distance function) that:

  (non-negativity)

Definition of metric space

A metric space is an ordered pair where is a set and is a metric on .

For shorthand, a metric space is usually written simply as once the metric has been defined or is understood.

Metric topology

A metric on a set induces a particular topology on called the metric topology. For any , let the open ball of radius around the point be defined as . Define the collection of subsets of (meaning that ) consisting of the empty set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \emptyset} and all sets of the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cup_{\gamma \in \Gamma}B_{r_{\gamma}}(x_{\gamma}),}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma\,} is an arbitrary index set (can be uncountable) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_{\gamma}>0\,} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{\gamma} \in X} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma \in \Gamma} . Then the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{O}} satisfies all the requirements to be a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\,} and is said to be the topology induced by the metric Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\,} . Any topology induced by a metric is said to be a metric topology.

Examples

  1. The "canonical" example of a metric space, and indeed what motivated the general definition of such a space, is the Euclidean space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}^n} endowed with the Euclidean distance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_E\,} defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_E(x,y)=\sqrt{\sum_{k=1}^{n}|x_k-y_k|^2}} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,y \in \mathbb{R}^n } .
  2. Consider the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C[a,b]\,} of all real valued continuous functions on the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]\subset \mathbb{R}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a<b\,} . Define the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d:C[a,b] \times C[a,b] \rightarrow \mathbb{R}} by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d(f,g)=\max_{x \in [a,b]}|f(x)-g(x)| } for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f,g \in C[a,b]} . This function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d\,} is a metric on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C[a,b]\,} and induces a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C[a,b]\,} often known as the norm topology or uniform topology.
  3. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\,} be any nonempty set. The discrete metric on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X\,} is defined as if and otherwise. In this case the induced topology is the discrete topology, in which every set is open.

Mappings

A mapping f from a metric space (X,d) to another (Y,e) is an isometry if it is distance-preserving: that is

A mapping f from a metric space (X,d) to another (Y,e) is continuous at x in X if for all real ε > 0 there exists δ > 0 such that

and continuous if it is continuous at every point of X.

If we let denote the open ball of radius r round x in X, and similarly denote the open ball of radius r round y in Y, we can express these conditions in terms of the pull-back

See also

Topology

Topological space

Normed space


References

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980

Retrieved from "https://citizendium.org/wiki/index.php?title=Metric_space&oldid=980754"