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]]

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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.

See also