Nonarchimedean Functional Analysis

This book grew out of a course which I gave during the winter term 1997/98 at the Universitat Munster. The course covered the material which here is presented in the first three chapters. The fourth more advanced chapter was added to give the reader a rather complete tour through all the important aspects of the theory of locally convex vector spaces over nonarchimedean fields. There is one serious restriction, though, which seemed inevitable to me in the interest of a clear presentation. In its deeper aspects the theory depends very much on the field being spherically complete or not. To give a drastic example, if the field is not spherically complete then there exist nonzero locally convex vector spaces which do not have a single nonzero continuous linear form. Although much progress has been made to overcome this problem a really nice and complete theory which to a large extent is analogous to classical functional analysis can only exist over spherically complete field8. I therefore allowed myself to restrict to this case whenever a conceptual clarity resulted. Although I hope that thi8 text will also be useful to the experts as a reference my own motivation for giving that course and writing this book was different. I had the reader in mind who wants to use locally convex vector spaces in the applications and needs a text to quickly gra8p this theory.

Covers all the important aspects of the theory of locally convex vector spaces over nonarchimedean fields Gives the foundations of the theory and also develops the more advanced topics Concise introduction for the researcher and the graduate student who want to apply this theory Includes supplementary material: sn.pub/extras

Klappentext

The present book is a self-contained text which leads the reader through all the important aspects of the theory of locally convex vector spaces over nonarchimedean fields. One can observe an increasing interest in methods from nonarchimedean functional analysis, particularly in number theory and in the representation theory of p-adic reductive groups. The book gives a concise and clear account of this theory, it carefully lays the foundations and also develops the more advanced topics. Although the book will be a valuable reference work for experts in the field, it is mainly intended as a streamlined but detailed introduction for researchers and graduate students who wish to apply these methods in different areas.

Inhalt
I. Foundations.- Nonarchimedean Fields; Seminorms; Normed Vector Spaces; Locally Convex Vector Spaces; Constructions and Examples; Spaces of Continuous Linear Maps; Completeness; Fréchet Spaces; the Dual Space. - II. The Structure of Banach Spaces.- Structure theorems; Non-Reflexivity.- III. Duality Theory.- C-Compact and Compactoid Submodules; Polarity; Admissible Topologies; Reflexivity; Compact Limits.- IV. Nuclear Maps and Spaces.- Topological Tensor Products; Completely Continuous Maps; Nuclear Spaces; Nuclear Maps; Traces; Fredholm Theory.- References.- Index, Notations.