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.
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." (Dinamérico 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)
