Spherically complete field
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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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