Nonarchimedean Functional Analysis

From the reviews of the first edition: "It is the first textbook seriously covering locally convex theory over K, so ... it is most welcome. ... the book is self-contained, complete with all proofs, and therefore attractive also to those who are not acquainted with the above area. ... The book is well-written, with care for details. Recommended." (W.H. Schikhof, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 106 (1), 2004) "The book under review is a self-contained text concerning the theory of locally convex spaces over non-Archimedean fields. ... The book is carefully written and incorporates for the first time results that have only appeared in papers. It will be a valuable reference work either for specialists or for non-specialists in the field." (Dinamerico P. Pombo, Jr., Mathematical Reviews, Issue 2003 a) "Functional analysis over nonarchimedean fields has become an area of growing interest ... . In the present book the author gives a concise and clear account of this theory, carefully lays the foundations, and also develops the more advanced topics. ... This book gives a streamlined introduction for researchers and graduate students who want to apply these methods to other areas, and it would probably also provide a valuable reference source for researchers in the field." (Anton Deitmar, Bulletin of the London Mathematical Society, Vol. 34, 2002) "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. ... 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 streamlined but detailed introduction for researchers and graduate students ... ." (L'Enseignement Mathematique, Vol. 48 (1-2), 2002)

• 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.