Spherically complete field

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]

${\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots \Rightarrow \bigcap _{n\in {\mathbf {N} }}B_{n}\neq \emptyset .}$

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 Q_p of p-adic numbers, and any of their finite extensions.
• Every spherically complete field is complete. On the other hand, C_p, the completion of the algebraic closure of Q_p, is not spherically complete.^[3]
• Any field of Hahn series is spherically complete.

References

1. 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.
2. Schneider, P. (2002). Nonarchimedean functional analysis. Springer monographs in mathematics. Berlin ; New York: Springer. ISBN 978-3-540-42533-5.
3. Robert, Alain M. (2000-05-31). A Course in p-adic Analysis. Springer Science & Business Media. p. 129. ISBN 978-0-387-98669-2.