# 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:

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

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.

## References

1. ^ Robert, p. 143

Schneider, Peter (2001). Nonarchimedean Functional Analysis. Springer. ISBN 3-540-42533-0.