Namespaces
Variants
Actions

Difference between revisions of "Cauchy sequence"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
Line 1: Line 1:
The same as a [[Fundamental sequence|fundamental sequence]].
+
{{MSC|40A05|54E35}}
 +
{{TEX|done}}
 +
 
 +
''Cauchy sequence, of points in a metric space $(X,d)$''
 +
 
 +
A  sequence $\{p_i\}$ of elements in a [[Metric space|metric space]] $(X,d)$ such that for any $\varepsilon > 0$ there is a number $N$ such that
 +
\[
 +
d (x_, x_m) < \varepsilon \qquad \forall m,n\geq N\, .
 +
\]
 +
The latter is also called Cauchy condition. A convergent sequence is always necessarily a Cauchy sequence. However the converse is not necessarily true. A metric space with the property that any Cauchy sequence has a limit is called [[Complete metric space|complete]], see also [[Cauchy criteria]])
 +
 
 +
The concept of Cauchy sequence can be generalized to Cauchy nets (see also [[Net]]; [[Net (of sets in a topological space)]], [[Generalized sequence]] and [[Cauchy filter]]) in a [[Uniform space|uniform space]]. Let $X$ be a uniform space  with uniformity $\mathcal{U}$. A net $\{x_\alpha, \alpha \in A\}$ (where $A$ is a [[Directed set|directed set]] of elements $x_\alpha \in X$, is called a Cauchy net if for every element  $U\in \mathcal{U}$ there is an index $\alpha_0 \in A$ such that for all
 +
\[
 +
(x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, .
 +
\]
 +
 
 +
====References====
 +
{|
 +
|-
 +
|valign="top"|{{Ref|Al}}|| P.S.  Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine  Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from  Russian)
 +
|-
 +
|valign="top"|{{Ref|Du}}|| J. Dugundji,  "Topology" , Allyn &amp; Bacon  (1966)  {{MR|0193606}} {{ZBL|0144.21501}}
 +
|-
 +
|valign="top"|{{Ref|Ke}}|| J.L. Kelley,    "General topology" , Springer  (1975)
 +
|-
 +
|valign="top"|{{Ref|KF}}|| A.N. Kolmogorov,    S.V. Fomin,  "Elements of the theory of functions and functional  analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from  Russian)
 +
|-
 +
|}

Revision as of 10:10, 9 December 2013

2020 Mathematics Subject Classification: Primary: 40A05 Secondary: 54E35 [MSN][ZBL]

Cauchy sequence, of points in a metric space $(X,d)$

A sequence $\{p_i\}$ of elements in a metric space $(X,d)$ such that for any $\varepsilon > 0$ there is a number $N$ such that \[ d (x_, x_m) < \varepsilon \qquad \forall m,n\geq N\, . \] The latter is also called Cauchy condition. A convergent sequence is always necessarily a Cauchy sequence. However the converse is not necessarily true. A metric space with the property that any Cauchy sequence has a limit is called complete, see also Cauchy criteria)

The concept of Cauchy sequence can be generalized to Cauchy nets (see also Net; Net (of sets in a topological space), Generalized sequence and Cauchy filter) in a uniform space. Let $X$ be a uniform space with uniformity $\mathcal{U}$. A net $\{x_\alpha, \alpha \in A\}$ (where $A$ is a directed set of elements $x_\alpha \in X$, is called a Cauchy net if for every element $U\in \mathcal{U}$ there is an index $\alpha_0 \in A$ such that for all \[ (x_\alpha, x_\beta)\in U \qquad \forall \alpha, \beta \geq \alpha_0\, . \]

References

[Al] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[Du] J. Dugundji, "Topology" , Allyn & Bacon (1966) MR0193606 Zbl 0144.21501
[Ke] J.L. Kelley, "General topology" , Springer (1975)
[KF] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
How to Cite This Entry:
Cauchy sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cauchy_sequence&oldid=30874