Samsung NX200

2013-12-05T05:42:42Z

121.97.248.16: /* Changes from NX100 */

In mathematics, a [[Field (mathematics)|field]] ''K'' with an [[Absolute_value#Fields|absolute value]] is called '''spherically complete''' if the [[Intersection (set theory)|intersection]] of every [[decreasing sequence]] of [[ball]]s (in the sense of the metric induced by the absolute value) is nonempty:
:<math>B_1\supseteq B_2\supseteq \cdots \Rightarrow\bigcap_{n\in {\mathbf N}} B_n\neq \empty.</math>
The definition can be adapted also to a field ''K'' with a [[Valuation (algebra)|valuation]] ''v'' taking values in an arbitrary ordered abelian group: (''K'',''v'') is spherically complete if every collection of balls that is totally ordered by inclusion has a nonempty intersection.

Spherically complete fields are important in [[archimedean property|nonarchimedean]] [[functional analysis]], since many results analogous to theorems of classical functional analysis require the base field to be spherically complete.

==Examples==
*Any [[locally compact]] field is spherically complete. This includes, in particular, the fields '''Q'''''p'' of [[p-adic number]]s, and any of their finite extensions.
*On the other hand, '''C'''''p'', the [[complete metric space|completion]] of the [[algebraic closure]] of '''Q'''''p'', is not spherically complete.
*Any field of [[Hahn series]] is spherically complete.

==References==
{{cite book |title=Nonarchimedean Functional Analysis
|last=Schneider
|first=Peter
|year=2001
|publisher=Springer
|isbn=3-540-42533-0}}

[[Category:Algebra]]
[[Category:Functional analysis]]

{{mathanalysis-stub}}

formulasearchengine - User contributions [en]

Samsung NX200