# 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:

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.

## 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. - On the other hand,
**C**_{p}, the completion of the algebraic closure of**Q**_{p}, is not spherically complete.[1] - Any field of Hahn series is spherically complete.

## References

- Robert, p. 143

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

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