Nonarchimedean Functional Analysis

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Springer Science & Business Media, Nov 20, 2001 - Mathematics - 156 pages
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.
 

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Contents

Chapter I Foundations
1
Nonarchimedean Fields
2
Seminorms
6
Normed Vector Spaces
8
Locally Convex Vector Spaces
13
Constructions and Examples
19
Spaces of Continuous Linear Maps
27
Completeness
35
Admissible Topologies
83
Reflexivity
86
Compact Limits
89
Chapter IV Nuclear Maps and Spaces
101
Topological Tensor Products
102
Completely Continuous Maps
113
Nuclear Spaces
119
Nuclear Maps
125

Frechet Spaces
45
The Dual Space
50
Chapter II The Structure of Banach Spaces
59
Nonreflexivity
64
Chapter III Duality Theory
67
cCompact and Compactoid Submodules
68
Polarity
76
Traces
136
Fredholm Theory
140
References
151
Notations
153
Index
155
Copyright

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Page 152 - Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366-383.
Page 152 - WH Schikhof. Locally convex spaces over non-spherically complete valued fields I, II. Bull. Soc. Math. Belg.
Page 151 - GRUSON, L., Theorie de Fredholm p-adique, Bull. Soc. Math. France 94, 67-95 (1966). [4] GRUSON, L., Cathégories d'espaces de Banach ul tramé triques, Bull.
Page 152 - Une notion de compacite dans la theorie des espaces vectoriels topologiques, Indagationes Math. 27. 182-189 (1965) van der Put M., Reflexive non-archimedean Banach spaces, Indagationes Math.

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