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''distance on a set $X$ ''
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{{TEX|done}}
  
A function $\rho$ with non-negative real values, defined on the Cartesian product $X\times X$ and satisfying for any $x, y\in X$ the conditions:
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''distance on a [[set]] $X$ ''
  
1) $\rho(x,y)=0$ if and only if $x = y$ (the identity axiom);
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A function $\rho$ with non-negative real values, defined on the [[Cartesian product]] $X\times X$ and satisfying for any $x, y\in X$ the conditions:
 +
# $\rho(x,y)=0$ if and only if $x = y$ (the identity axiom);
 +
# $\rho(x,y) + \rho(y,z) \geq \rho(x,z)$ (the triangle axiom);
 +
# $\rho(x,y) = \rho(y,x)$ (the symmetry axiom).
  
2) $\rho(x,y) + \rho(y,z) \geq \rho(x,z)$ (the triangle axiom);
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A set $X$ on which it is possible to introduce a metric is called metrizable (cf. [[Metrizable space]]). A set $X$ provided with a metric is called a [[metric space]].
  
3) $\rho(x,y) = \rho(y,x)$ (the symmetry axiom).
 
  
A set $X$ on which it is possible to introduce a metric is called metrizable (cf. [[Metrizable space|Metrizable space]]). A set $X$ provided with a metric is called a [[Metric space|metric space]].
 
  
 
===Examples.===
 
===Examples.===
 
  
 
1) On any set there is the discrete metric
 
1) On any set there is the discrete metric
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2) In the space $\mathbb R^n$ various metrics are possible, among them are:
 
2) In the space $\mathbb R^n$ various metrics are possible, among them are:
 +
\begin{equation}
 +
\rho(x,y) = \sqrt{\sum(x_i-y_i)^2};
 +
\end{equation}
 +
\begin{equation}
 +
\rho(x,y)=\sup\limits_i|x_i-y_i|;
 +
\end{equation}
 +
\begin{equation}
 +
\rho(x,y)=\sum|x_i-y_i|;
 +
\end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362013.png" /></td> </tr></table>
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here $\{x_i\}, \{y_i\} \in \mathbb{R}^n$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362014.png" /></td> </tr></table>
+
3) In a [[Riemannian space]] a metric is defined by a [[metric tensor]], or a quadratic differential form (in some sense, this is an analogue of the first metric of example 2)). For a generalization of metrics of this type see [[Finsler space]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362015.png" /></td> </tr></table>
+
4) In function spaces on a (countably) [[compact space]] $X$ there are also various metrics; for example, the uniform metric
 +
\begin{equation}
 +
\rho(f,g)=\sup\limits_{x\in X}|f(x)-g(x)|
 +
\end{equation}
 +
(an analogue of the second metric of example 2)), and the integral metric
 +
\begin{equation}
 +
\rho(f,g)=\int\limits_X|f-g|\, dx.
 +
\end{equation}
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362016.png" />.
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5) In [[normed space]]s over $\mathbb R$ a metric is defined by the norm $\|\cdot\|$:
 +
\begin{equation}
 +
\rho(x,y) = \|x-y\|.
 +
\end{equation}
  
3) In a Riemannian space a metric is defined by a [[Metric tensor|metric tensor]], or a quadratic differential form (in some sense, this is an analogue of the first metric of example 2)). For a generalization of metrics of this type see [[Finsler space|Finsler space]].
+
6) In the space of closed subsets of a metric space there is the [[Hausdorff metric]].
  
4) In function spaces on a (countably) compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362017.png" /> there are also various metrics; for example, the uniform metric
+
If, instead of 1), one requires only:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362018.png" /></td> </tr></table>
+
1') $\rho(x,y)=0$ if $x=y$ (so that from $\rho(x,y)=0$ it does not always follows that $x=y$), the function $\rho$ is called a [[pseudo-metric]] <ref name="Kelley" /><ref name="Kuratowski" />, or finite écart <ref name="Bourbaki" />.
  
(an analogue of the second metric of example 2)), and the integral metric
+
A metric (and even a pseudo-metric) makes the definition of a number of additional structures on the set $X$ possible. First of all a topology (see [[Topological space]]), and in addition a uniformity (see [[Uniform space]]) or a proximity (see [[Proximity space]]) structure. The term metric is also used to denote more general notions which do not have all the properties 1)–3); such are, for example, an [[indefinite metric]], a [[symmetry on a set]], etc.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362019.png" /></td> </tr></table>
 
  
5) In normed spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362020.png" /> a metric is defined by the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362021.png" />:
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362022.png" /></td> </tr></table>
+
====Comments====
  
6) In the space of closed subsets of a metric space there is the [[Hausdorff metric|Hausdorff metric]].
+
Potentially, any metric space $(X,\rho)$ has a second metric $\sigma \geq \rho$ naturally associated: the intrinsic or [[internal metric]]. Potentially, because the definition may give $\sigma(x,y)=\infty$ for some pairs of points $x, y$. One defines the length (which may be $\infty$) of a continuous path $f:[0,1]\to X$ by $L(f)=\lim\limits_{\epsilon\to 0}\sup L_{\epsilon}(f)$, where $L_{\epsilon}(f)$ is the infimum of all finite sums $\sum \rho(x_i,x_{i+1})$ with $\{x_i\}$ a finite subset of $[0,1]$ which is an $\epsilon$-net (cf. [[Metric space]]) and is listed in the natural order. Then $\sigma(x,y)$ is the infimum of the lengths of paths $f$ with $f(0)=x$, $f(1)=y$, but $\sigma(x,y)=\infty$ if there is no such path of finite length.
  
If, instead of 1), one requires only:
+
No reasonable topological restriction on $(X,\rho)$ suffices to guarantee that the intrinsic  "metric"  (or écart) $\sigma$ will be finite-valued. If $\sigma$ is finite-valued, suitable compactness conditions will assure that minimum-length paths, i.e. paths from $x$ to $y$ of length $\sigma(x,y)$, exist. When every pair of points $x, y$ is joined by a path (non-unique, in general) of length $\sigma(x,y)$, the metric is often called convex. (This is much weaker than the surface theorists' [[Convex metric|convex metric]].) The main theorem in this area is that every locally connected metric [[continuum]] admits a convex metric <ref name="Bing" /><ref name="Moïse" />.
  
1') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362024.png" /> (so that from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362025.png" /> it does not always follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362026.png" />), the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362027.png" /> is called a pseudo-metric [[#References|[2]]], [[#References|[3]]], or finite écart [[#References|[4]]].
 
  
A metric (and even a pseudo-metric) makes the definition of a number of additional structures on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362028.png" /> possible. First of all a topology (see [[Topological space|Topological space]]), and in addition a uniformity (see [[Uniform space|Uniform space]]) or a proximity (see [[Proximity space|Proximity space]]) structure. The term metric is also used to denote more general notions which do not have all the properties 1)–3); such are, for example, an [[Indefinite metric|indefinite metric]], a [[Symmetry on a set|symmetry on a set]], etc.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K. Kuratowski,  "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966)  (Translated from French)</TD></TR></table>
 
 
 
 
====Comments====
 
Potentially, any metric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362029.png" /> has a second metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362030.png" /> naturally associated: the intrinsic or [[Internal metric|internal metric]]. Potentially, because the definition may give <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362031.png" /> for some pairs of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362032.png" />. One defines the length (which may be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362033.png" />) of a continuous path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362034.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362036.png" /> is the infimum of all finite sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362037.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362038.png" /> a finite subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362039.png" /> which is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362040.png" />-net (cf. [[Metric space|Metric space]]) and is listed in the natural order. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362041.png" /> is the infimum of the lengths of paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362042.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362044.png" />, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362045.png" /> if there is no such path of finite length.
 
  
No reasonable topological restriction on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362046.png" /> suffices to guarantee that the intrinsic  "metric"  (or écart) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362047.png" /> will be finite-valued. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362048.png" /> is finite-valued, suitable compactness conditions will assure that minimum-length paths, i.e. paths from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362049.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362050.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362051.png" />, exist. When every pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362052.png" /> is joined by a path (non-unique, in general) of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063620/m06362053.png" />, the metric is often called convex. (This is much weaker than the surface theorists' [[Convex metric|convex metric]].) The main theorem in this area is that every locally connected metric [[Continuum|continuum]] admits a convex metric [[#References|[a1]]], [[#References|[a2]]].
+
<references>
 
+
<ref name="Kelley">J.L. Kelley,  "General topology" , Springer (1975)</ref>
====References====
+
<ref name="Kuratowski">K. Kuratowski,  "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</ref>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.H. Bing,  "Partitioning a set"  ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 1101–1110</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.E. Moïse,  "Grille decomposition and convexification"  ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 1111–1121</TD></TR></table>
+
<ref name="Bourbaki">N. Bourbaki,  "Elements of mathematics. General topology" , Addison-Wesley  (1966) (Translated from French)</ref>
 +
<ref name="Bing">R.H. Bing,  "Partitioning a set"  ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 1101–1110</ref>
 +
<ref name="Moïse">E.E. Moïse,  "Grille decomposition and convexification"  ''Bull. Amer. Math. Soc.'' , '''55'''  (1949)  pp. 1111–1121</ref>
 +
</references>
 +
<ol start="6">
 +
<li>P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</li>
 +
</ol>

Latest revision as of 11:16, 7 June 2016


distance on a set $X$

A function $\rho$ with non-negative real values, defined on the Cartesian product $X\times X$ and satisfying for any $x, y\in X$ the conditions:

  1. $\rho(x,y)=0$ if and only if $x = y$ (the identity axiom);
  2. $\rho(x,y) + \rho(y,z) \geq \rho(x,z)$ (the triangle axiom);
  3. $\rho(x,y) = \rho(y,x)$ (the symmetry axiom).

A set $X$ on which it is possible to introduce a metric is called metrizable (cf. Metrizable space). A set $X$ provided with a metric is called a metric space.


Examples.

1) On any set there is the discrete metric \begin{equation} \rho(x,y) = 0 \text{ if } x=y \quad \text{and} \quad \rho(x,y) = 1 \text{ if } x\ne y. \end{equation}

2) In the space $\mathbb R^n$ various metrics are possible, among them are: \begin{equation} \rho(x,y) = \sqrt{\sum(x_i-y_i)^2}; \end{equation} \begin{equation} \rho(x,y)=\sup\limits_i|x_i-y_i|; \end{equation} \begin{equation} \rho(x,y)=\sum|x_i-y_i|; \end{equation}

here $\{x_i\}, \{y_i\} \in \mathbb{R}^n$.

3) In a Riemannian space a metric is defined by a metric tensor, or a quadratic differential form (in some sense, this is an analogue of the first metric of example 2)). For a generalization of metrics of this type see Finsler space.

4) In function spaces on a (countably) compact space $X$ there are also various metrics; for example, the uniform metric \begin{equation} \rho(f,g)=\sup\limits_{x\in X}|f(x)-g(x)| \end{equation} (an analogue of the second metric of example 2)), and the integral metric \begin{equation} \rho(f,g)=\int\limits_X|f-g|\, dx. \end{equation}

5) In normed spaces over $\mathbb R$ a metric is defined by the norm $\|\cdot\|$: \begin{equation} \rho(x,y) = \|x-y\|. \end{equation}

6) In the space of closed subsets of a metric space there is the Hausdorff metric.

If, instead of 1), one requires only:

1') $\rho(x,y)=0$ if $x=y$ (so that from $\rho(x,y)=0$ it does not always follows that $x=y$), the function $\rho$ is called a pseudo-metric [1][2], or finite écart [3].

A metric (and even a pseudo-metric) makes the definition of a number of additional structures on the set $X$ possible. First of all a topology (see Topological space), and in addition a uniformity (see Uniform space) or a proximity (see Proximity space) structure. The term metric is also used to denote more general notions which do not have all the properties 1)–3); such are, for example, an indefinite metric, a symmetry on a set, etc.


Comments

Potentially, any metric space $(X,\rho)$ has a second metric $\sigma \geq \rho$ naturally associated: the intrinsic or internal metric. Potentially, because the definition may give $\sigma(x,y)=\infty$ for some pairs of points $x, y$. One defines the length (which may be $\infty$) of a continuous path $f:[0,1]\to X$ by $L(f)=\lim\limits_{\epsilon\to 0}\sup L_{\epsilon}(f)$, where $L_{\epsilon}(f)$ is the infimum of all finite sums $\sum \rho(x_i,x_{i+1})$ with $\{x_i\}$ a finite subset of $[0,1]$ which is an $\epsilon$-net (cf. Metric space) and is listed in the natural order. Then $\sigma(x,y)$ is the infimum of the lengths of paths $f$ with $f(0)=x$, $f(1)=y$, but $\sigma(x,y)=\infty$ if there is no such path of finite length.

No reasonable topological restriction on $(X,\rho)$ suffices to guarantee that the intrinsic "metric" (or écart) $\sigma$ will be finite-valued. If $\sigma$ is finite-valued, suitable compactness conditions will assure that minimum-length paths, i.e. paths from $x$ to $y$ of length $\sigma(x,y)$, exist. When every pair of points $x, y$ is joined by a path (non-unique, in general) of length $\sigma(x,y)$, the metric is often called convex. (This is much weaker than the surface theorists' convex metric.) The main theorem in this area is that every locally connected metric continuum admits a convex metric [4][5].


References

  1. J.L. Kelley, "General topology" , Springer (1975)
  2. K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)
  3. N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French)
  4. R.H. Bing, "Partitioning a set" Bull. Amer. Math. Soc. , 55 (1949) pp. 1101–1110
  5. E.E. Moïse, "Grille decomposition and convexification" Bull. Amer. Math. Soc. , 55 (1949) pp. 1111–1121
  1. P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
How to Cite This Entry:
Metric. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Metric&oldid=29395
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article