Metric space: Difference between revisions

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#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>.  
#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)| </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''.
#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 so called ''discrete 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>


== See also ==
== See also ==

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In mathematics, a metric space is, roughly speaking, an abstract mathematical structure that generalizes the notion of a 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} 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 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 an arbitrary set. A 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\,} 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 a 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: X \times X \rightarrow \mathbb{R}}   with the following properties:

  1. 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(x_2,x_1)=d(x_1,x_2) \quad \forall x_1,x_2 \in X}   (symmetry)
  2. 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(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}   (triangular inequality)
  3. 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(x_1,x_2)=0\ \Leftrightarrow \ x_1=x_2\,}

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

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(x_1,x_2) \geq 0 \quad \forall x_1,x_2 \in X}   (non-negativity)

Definition of metric space

A metric space is an ordered pair 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,d)\,} 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 X\,} is a set 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 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 X\,} .

For shorthand, a metric 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 (X,d)\,} is usually written simply as 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\,} once 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\,} has been defined or is understood.

Metric topology

A metric on a 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 X\,} induces a particular 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\,} called the metric topology. For any 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 \in X } , let the open ball 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 B_r(x)\,} of radius 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>0\,} around the point 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 defined as 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 B_r(x)=\{y \in X \mid d(y,x)<r\}} . Define the collection 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}} of subsets of 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\,} (meaning that 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 \in \mathcal{O} \Rightarrow A \subset X } ) 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

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

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