Banach spaces over nonarchimedian valued fields
by
Wim H. Schikhof
University of Nijmegen, The Netherlands

In this survey note we present the state of the art on the theory of Banach and Hilbert spaces over complete valued scalar fields that are not isomorphic to R or C. We consider `classical' theorems such as the Hahn-Banach, Closed Range, Riesz Representation, Eberlein- Smulian, and Krein-Milman theorems and discuss whether or not they remain valid in this new context; in some cases there are strong negations, in others strong improvements. Several new concepts are introduced (e.g., norm orthogonality, spherical completeness, compactoidity, modules over valuation rings) leading to results that do not have-or at least have less important-counterparts in the classical theory.

Date received: May 26, 1999