Spherically complete field

Mathematical term From Wikipedia, the free encyclopedia

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]


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]

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