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.
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Chapter I Foundations
Normed Vector Spaces
Locally Convex Vector Spaces
Constructions and Examples
Spaces of Continuous Linear Maps
Chapter IV Nuclear Maps and Spaces
Topological Tensor Products
Completely Continuous Maps
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A C V According to Lemma According to Prop assume B C V Banach space barrelled bijection bornological bounded and c-compact bounded o-submodule bounded open lattice bounded subset canonical map Cauchy net closure compact map compactoid continuous linear form continuous linear map continuous seminorm converges convex A'-vector spaces convex vector spaces Corollary denote duality map equicontinuous family of lattices family of seminorms final topology finite rank operator follows from Prop Frechet space hand side Hausdorff hence implies injective K-vector Let L C V lim Vn locally convex A'-vector locally convex topology locally convex vector map f nonarchimedean normed vector space o-module open lattice L C V pL(v projective tensor product Proposition pseudo-polar quasi-complete quotient reflexive Remark resp scalar seminorm q spherically complete subspace topology Suppose surjective tensor product topology topological isomorphism vector subspace weak topology zero vector
Page 152 - Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366-383.
Page 151 - GRUSON, L., Theorie de Fredholm p-adique, Bull. Soc. Math. France 94, 67-95 (1966).  GRUSON, L., Cathégories d'espaces de Banach ul tramé triques, Bull.