Introduction to Tensor Products of Banach Spaces

This book is intended as an introduction to the theory of tensor products of Banach spaces. The prerequisites for reading the book are a first course in Functional Analysis and in Measure Theory, as far as the Radon-Nikodym theorem. The book is entirely self-contained and two appendices give addi­ tional material on Banach Spaces and Measure Theory that may be unfamil­ iar to the beginner. No knowledge of tensor products is assumed. Our viewpoint is that tensor products are a natural and productive way to understand many of the themes of modern Banach space theory and that "tensorial thinking" yields insights into many otherwise mysterious phenom­ ena. We hope to convince the reader of the validity of this belief. We begin in Chapter 1 with a treatment of the purely algebraic theory of tensor products of vector spaces. We emphasize the use of the tensor product as a linearizing tool and we explain the use of tensor products in the duality theory of spaces of operators in finite dimensions. The ideas developed here, though simple, are fundamental for the rest of the book.

From the reviews:

"The book under review is intended to serve as an introduction to the theory of tensor products of Banach spaces. ... it is a most welcome addition to the existing literature and appears to be well-suited as a guide and as a textbook in lectures, seminars, etc., for students ... . Each chapter is accompanied by a set of exercises. ... The book is very carefully written and edited. The text makes very good reading ... ." (Hans Jarchow, Zentralblatt MATH, Vol. 1090 (16), 2006)

"This book provides a self-contained introduction to tensor products of Banach spaces. ... Each chapter is accompanied by exercises with illustrate mathematical concepts and propositions introduced there. The more or less straightforward solution of most of these exercises leads the reader to a better understanding of the material. ... It is accessible, clearly organized, and well written. It can be highly recommended to graduate students and mathematicians interested in functional analysis." (K. –D Kürsten, Zeitschrift für Analysis und ihre Anwendungen, Vol. 24 (4), 2005)

"The author sets out to give an accessible account of some chosen topics from the theory of tensor products for Banach spaces ... . The result is a very well written, beautiful book ... presenting a coherent, originally arranged and presented overview of various classical results. The book is self-contained and presented in a way that it at the same time is an accessible text for graduate students, as well as a very good basic reference for researchers ... ." (Mark Sioen, Simon Stevin Bulletin, Vol. 11 (2), 2004)

"This book is a good source to learn the basic notions of tensor products ... . The book can be recommended to everybody who is interested in the theory of tensor products and their applications, but first of all, to beginners in this field." (Endre Durszt, Acta Scientiarum Mathematicarum, Vol. 69, 2003)

"This book provides a self-contained introduction to the theory of tensor products on Banach spaces. Being the first book with a clear focus not only on graduate students in analysis, but also researchers who wish to become acquainted with the elementary facts of this area, it is a welcome addition to the literature. ... The book contains a large number of interesting exercises." (Andreas Defant, Mathematical Reviews, Issue 2003 f)

"The aim of the present book is to give a thorough and relatively short introduction to tensor products of Banach spaces ... and bringing the reader to the frontier of current research in the area. ... The book is an excellent introduction to the theory of tensor products and it is highly recommended to graduate students in analysis and to researchers in other areas needing results from this field." (S. Cobzas, Studia Universitatis Babes-Bolyai, Vol. XLVII (4), 2002)