Complete metric space: Difference between revisions
Jump to navigation
Jump to search
imported>Larry Sanger m (Completeness moved to Completeness (mathematics): It's the name of a different property, in logic) |
imported>Subpagination Bot m (Add {{subpages}} and remove any categories (details)) |
||
| Line 1: | Line 1: | ||
{{subpages}} | |||
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." | 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." | ||
| Line 8: | Line 10: | ||
[[Hilbert space]] | [[Hilbert space]] | ||
Revision as of 12:26, 27 January 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 .
