The Weierstrass theorem in fields with valuations

Abstract

irving kaplanskyIn [2, Theorem 32 ]2 the author showed that an analogue of the Weierstrass-Stone theorem holds in topological rings having ideal neighborhoods of 0. Earlier, Dieudonne [l] had proved the Weierstrass-Stone theorem for the field of p-adic numbers.Now the field of p-adic numbers has an open subring (the £-adic integers) with ideal neighborhoods of 0. It seems plausible, therefore, to expect that the method of [2 ] will apply, provided one has a supplementary device for "getting into" the £-adic integers.This is in fact the case, and the result is applicable to any division ring with a valuation of rank one.'The requisite lemma reads as follows: