complete
A metric space $X$ is complete^{} if every Cauchy sequence^{} (http://planetmath.org/CauchySequence) in $X$ is a convergent sequence^{}.
Examples:
Cauchy sequence

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The space $\mathbb{Q}$ of rational numbers is not complete: the sequence $3$, $3.1$, $3.14$, $3.141$, $3.1415$, $3.14159$, $3.141592\mathrm{\dots}$ consisting of finite decimals converging to $\pi \in \mathbb{R}$ is a Cauchy sequence in $\mathbb{Q}$ that does not converge in $\mathbb{Q}$.

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The space $\mathbb{R}$ of real numbers is complete, as it is the completion of $\mathbb{Q}$ with respect to the standard metric (other completions, such as the $p$adic numbers, are also possible). More generally, the completion of any metric space is a complete metric space.

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Every Banach space^{} is complete. For example, the ${L}^{p}$–space of pintegrable functions is a complete metric space if $p\ge 1$.
Title  complete 

Canonical name  Complete 
Date of creation  20130322 11:55:11 
Last modified on  20130322 11:55:11 
Owner  djao (24) 
Last modified by  djao (24) 
Numerical id  12 
Author  djao (24) 
Entry type  Definition 
Classification  msc 54E50 