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# New Examples of Non-Archimedean Banach Spaces and Applications

Published:2011-06-29
Printed: Dec 2012
• C. Perez-Garcia,
Department of Mathematics, Facultad de Ciencias, Universidad de Cantabria, 39071 Santander, Spain
• W. H. Schikhof,
Department of Mathematics, Radboud University, 6525 ED Nijmegen, The Netherlands
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## Abstract

The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm, without non-zero vectors that are $\|\cdot\|$-orthogonal and such that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$ into a valued field.
 Keywords: non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases
 MSC Classifications: 46S10 - Functional analysis over fields other than ${\bf R}$ or ${\bf C}$ or the quaternions; non-Archimedean functional analysis [See also 12J25, 32P05] 12J25 - Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10]

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