{{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:<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>
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.<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== *Any locally compact field is spherically complete. This includes, in particular, the fields '''Q'''<sub>''p''</sub> of p-adic numbers, and any of their finite extensions. *Every spherically complete field is complete. On the other hand, '''C'''<sub>''p''</sub>, the 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.
==References== {{Reflist}} Category:Algebra Category:Functional analysis Category:Field theory
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