Spherically complete field explained

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]

B _{1\supseteq} B _{2\supseteq} … ⇒ c ap _{n\in} } B_n\neq \empty.

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.

Notes and References

1. Van der Put . Marius . 1969 . Espaces de Banach non archimédiens . Bulletin de la Société mathématique de France . 79 . 309–320 . 10.24033/bsmf.1685 . 0037-9484.
2. Book: Schneider, P. . Nonarchimedean functional analysis . 2002 . Springer . 978-3-540-42533-5 . Springer monographs in mathematics . Berlin ; New York.
3. Book: Robert, Alain M. . A Course in p-adic Analysis . 2000-05-31 . Springer Science & Business Media . 978-0-387-98669-2 . 129 . en.