Complete metric space: Difference between revisions
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==Formal definition== | ==Formal definition== | ||
Let ''X'' be a metric space with metric ''d''. Then ''X'' is complete if for every Cauchy sequence <math>x_1,x_2,\ldots \in X</math> there is an associated element <math>x \in X</math> such that <math>\mathop{\lim}_{n \rightarrow \infty} d(x_n,x)=0</math>. | Let ''X'' be a metric space with metric ''d''. Then ''X'' is complete if for every Cauchy sequence <math>x_1,x_2,\ldots \in X</math> there is an associated element <math>x \in X</math> such that <math>\mathop{\lim}_{n \rightarrow \infty} d(x_n,x)=0</math>. | ||
==Examples== | |||
* The real numbers '''R''', and more generally finite-dimensional [[Euclidean space]]s, with the usual metric are complete. | |||
==Completion== | |||
Every metric space ''X'' has a '''completion''' <math>\bar X</math> which is a complete metric space in which ''X'' is [[isometry|isometrically]] embedded as a [[Denseness|dense]] subspace. The completion has a universal property. | |||
===Examples=== | |||
* The real numbers '''R''' are the completion of the rational numbers '''Q''' with respect to the usual metric of absolute distance. | |||
==See also== | ==See also== | ||
[[Banach space]] | * [[Banach space]] | ||
* [[Hilbert space]] | |||
[[Hilbert space]] | |||
Revision as of 16:36, 1 November 2008
In mathematics, completeness is a property ascribed to a metric space in which every Cauchy sequence in that space is convergent. In other words, every Cauchy sequence in the metric space tends in the limit to a point which is again an element of that space. Hence the metric space is, in a sense, "complete."
Formal definition
Let X be a metric space with metric d. Then X is complete if for every Cauchy sequence there is an associated element such that .
Examples
- The real numbers R, and more generally finite-dimensional Euclidean spaces, with the usual metric are complete.
Completion
Every metric space X has a completion which is a complete metric space in which X is isometrically embedded as a dense subspace. The completion has a universal property.
Examples
- The real numbers R are the completion of the rational numbers Q with respect to the usual metric of absolute distance.