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A metric space  is complete if every Cauchy sequence in  is a convergent sequence.
Examples:
- The space
 of rational numbers is not complete: the sequence  ,  ,  ,  ,  ,  ,
 consisting of finite decimals converging to
 is a Cauchy sequence in
 that does not converge in
 .
- The space
 of real numbers is complete, as it is the completion of
 with respect to the standard metric (other completions, such as the  -adic numbers, are also possible). More generally, the completion of any metric space is a complete metric space.
- Every Banach space is complete. For example, the
 -space of p-integrable functions is a complete metric space if  .
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"complete" is owned by djao.
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(view preamble)
Cross-references: p-integrable functions, Banach space, numbers, standard metric, completion, real numbers, converge, Cauchy sequence, finite, sequence, rational numbers, convergent sequence, metric space
There are 46 references to this entry.
This is version 7 of complete, born on 2001-10-27, modified 2008-01-23.
Object id is 603, canonical name is Complete.
Accessed 15650 times total.
Classification:
| AMS MSC: | 54E50 (General topology :: Spaces with richer structures :: Complete metric spaces) |
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Pending Errata and Addenda
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