Bounded approximation properties in non‐archimedean Banach spaces

Abstract

Some non-archimedean bounded approximation properties are introduced and studied in this paper. As an application, an affirmative answer is given, for non-spherically complete base fields, to the following problem, posed in [13], p. 95: Does there exist an absolutely convex edged set B in a non-archimedean locally convex space such that its closure B is not edged?

The metric approximation property in non-archimedean normed spaces

“…This comparison, together with the one carried out in [8,Section 6], reveals sharp and interesting contrasts between the classical MAP and its non-archimedean counterpart.…”

“…The study of Grothendieck's approximation in non-archimedean Banach spaces was initiated in [8]. In the present paper we derive new results leading to improvements of [8] (see e.g.…”

“…It was shown in [8] that a large amount of non-archimedean normed spaces have the λ-BAP (λ > 1) and the MAP. In fact, the following result holds.…”

“…We recall the following fundamental notion from [8], where, for convenience (see Theorem 4.4 and Corollary 4.5), we include also non-complete spaces. Definition 1.1.…”