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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, standard metric, completion, real numbers, converge, Cauchy sequence, finite, sequence, rational numbers, convergent sequence, metric space
There are 47 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 15269 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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