Spherically complete field: Difference between revisions
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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 (mathematics)|balls]] (in the sense of the metric induced by the absolute value) is nonempty: | {{Short description|Mathematical term}} | ||
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 (mathematics)|balls]] (in the sense of the metric induced by the absolute value) is nonempty:<ref>{{Cite journal |last=Van der Put |first=Marius |date=1969 |title=Espaces de Banach non archimédiens |url=http://www.numdam.org/item?id=BSMF_1969__97__309_0 |journal=Bulletin de la Société Mathématique de France |volume=79 |pages=309–320 |doi=10.24033/bsmf.1685 |issn=0037-9484}}</ref> | |||
:<math>B_1\supseteq B_2\supseteq \cdots \Rightarrow\bigcap_{n\in {\mathbf N}} B_n\neq \empty.</math> | :<math>B_1\supseteq B_2\supseteq \cdots \Rightarrow\bigcap_{n\in {\mathbf N}} B_n\neq \empty.</math> | ||
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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. | 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|functional analysis]], since many results analogous to theorems of classical functional analysis require the base field to be spherically complete. | Spherically complete fields are important in [[Archimedean property|nonarchimedean]] [[Functional analysis|functional analysis]], since many results analogous to theorems of classical functional analysis require the base field to be spherically complete.<ref>{{Cite book |last=Schneider |first=P. |title=Nonarchimedean functional analysis |date=2002 |publisher=Springer |isbn=978-3-540-42533-5 |series=Springer monographs in mathematics |location=Berlin ; New York}}</ref> | ||
==Examples== | ==Examples== | ||
*Any locally compact field is spherically complete. This includes, in particular, the fields '''Q'''<sub>''p''</sub> of [[P-adic number|p-adic number]]s, and any of their finite extensions. | *Any locally compact field is spherically complete. This includes, in particular, the fields '''Q'''<sub>''p''</sub> of [[P-adic number|p-adic number]]s, and any of their finite extensions. | ||
*Every spherically complete field is [[Complete metric space|complete]]. On the other hand, '''C'''<sub>''p''</sub>, the [[Complete metric space|completion]] of the algebraic closure of '''Q'''<sub>''p''</sub>, is not spherically complete.<ref>Robert | *Every spherically complete field is [[Complete metric space|complete]]. On the other hand, '''C'''<sub>''p''</sub>, the [[Complete metric space|completion]] of the algebraic closure of '''Q'''<sub>''p''</sub>, is not spherically complete.<ref>{{Cite book |last=Robert |first=Alain M. |url=https://books.google.com/books?id=H6sq_x2-DgoC |title=A Course in p-adic Analysis |date=2000-05-31 |publisher=Springer Science & Business Media |isbn=978-0-387-98669-2 |pages=129 |language=en}}</ref> | ||
*Any field of [[Hahn series]] is spherically complete. | *Any field of [[Hahn series]] is spherically complete. | ||
==References== | ==References== | ||
{{Reflist}} | {{Reflist}} | ||
[[Category:Algebra]] | [[Category:Algebra]] | ||
[[Category:Functional analysis]] | [[Category:Functional analysis]] | ||
[[Category:Field (mathematics)]] | |||
{{Sourceattribution|Spherically complete field}} | {{Sourceattribution|Spherically complete field}} | ||
Latest revision as of 04:59, 24 July 2025
Short description: Mathematical term
In mathematics, a field K with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty:[1]
- [math]\displaystyle{ 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 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 nonarchimedean functional analysis, since many results analogous to theorems of classical functional analysis require the base field to be spherically complete.[2]
Examples
- Any locally compact field is spherically complete. This includes, in particular, the fields Qp of p-adic numbers, and any of their finite extensions.
- Every spherically complete field is complete. On the other hand, Cp, the completion of the algebraic closure of Qp, is not spherically complete.[3]
- Any field of Hahn series is spherically complete.
References
- ↑ Van der Put, Marius (1969). "Espaces de Banach non archimédiens". Bulletin de la Société Mathématique de France 79: 309–320. doi:10.24033/bsmf.1685. ISSN 0037-9484. http://www.numdam.org/item?id=BSMF_1969__97__309_0.
- ↑ Schneider, P. (2002). Nonarchimedean functional analysis. Springer monographs in mathematics. Berlin ; New York: Springer. ISBN 978-3-540-42533-5.
- ↑ Robert, Alain M. (2000-05-31) (in en). A Course in p-adic Analysis. Springer Science & Business Media. pp. 129. ISBN 978-0-387-98669-2. https://books.google.com/books?id=H6sq_x2-DgoC.
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