Functional Analysis and Infinite-Dimensional Geometry

This book introduces the reader to the basic principles of functional analysis and to areas of Banach space theory that are close to nonlinear analysis and topology. In the first part, the book develops the classical theory, including weak topologies, locally convex spaces, Schauder bases, and compact operator theory. The presentation is self-contained, including many folklore results, and the proofs are accessible to students with the usual background in real analysis and topology. The second part covers topics in convexity and smoothness, finite representability, variational principles, homeomorphisms, weak compactness and more. Several results are published here for the first time in a monograph. The text can be used in graduate courses or for independent study. It includes a large number of exercises of different levels of difficulty, accompanied by hints. The book is also directed to young researchers in functional analysis and can serve as a reference book.This is an introduction to basic principles of functional analysis and to areas of Banach space theory close to nonlinear analysis and topology. The first part, which develops the classical theory, is self-contained and features a large number of exercises containing many important results. The second part covers selected topics in the theory of Banach spaces related to smoothness and topology. It is intended to be an introduction to and complement of existing books on the subject. This text may be used in graduate courses, for independent study, or as a reference book.

From the reviews:

"This is a substantial text containing up-to-date exposition and functional analysis from a Banach space point of view. It will be particularly useful for research investigation of nonlinear functional analysis and optimization...This book will stand as an important working text and reference and a significant guide for research students." (Mathematical Reviews)

"This book can be warmly recommended to everyone interested in functional analysis, and Banach space theory in particular. It serves also as a textbook in courses for students in probability, physics, or engineering. Graduate students and researchers surely will find a lot of material from the field, as well as a source of inspiration." (European Mathematical Society Newsletter, September, 2003)

"This is a substantial text containing an up-to-date exposition of functional analysis ... . It will be particularly useful for research investigation of nonlinear functional analysis and optimization. ... Each chapter ends with a remarkably weighty collection of exercises, many of which have useful hints at solutions appended to them ... . the reader is directed throughout to the ample collection of references. The book will stand as an important working text and reference and a significant guide for research students." (John R. Giles, Mathematical Reviews, Issue 2002 f)

"The sextet of authors have done a superb job in marshalling and presenting their material: the writing is crisp and authoritative and they take full advantage of recent simplifications in the proofs of certain results. The fulsome, up-to-date bibliography is accompanied by a marvellous collection of nearly 700 exercises (with integrated hints): for both learners and lecturers, this rich source of material alone is worth more than the cost of the book. ... I warmly commend this book ... ." (Nick Lord, The Mathematical Gazette, Vol. 87 (509), 2003)

"This book, which contains a vast amount of material, is intended as an introduction to linear function analysis ... . At the end of each chapter there is a wealth of beautiful applications and exercises ... . I would highly recommend this book to anyone interested in the study of Banach spaces ... . I think it would be fair to say that if one knew half of the material contained in this book, then one would know quite a lot." (Warren Moors, The Australian Mathematical Society Gazette, Vol. 29 (5), 2002)

"This book is based on graduate courses taught at the university of Alberta in Edmonton. It is intended as an introduction to linear functional analysis and to some parts of infinite-dimensional Banach space theory. It is full of facts, theorems, corollaries; along with a large number of exercises with detailed hints for their solution. ... The authors have accomplished a text which is easily readable and as self-contained as possible. A very excellent book for the topics covered." (Joe Howard, Zentralblatt MATH, Vol. 981, 2002)

"By its organization, the book can be used as a textbook for various types of courses in functional analysis ... . Besides classical material, the book contains also some recent and more specialized results ... . The book contains a large number of exercises with detailed hints, completing the main text with many important results. The book is a valuable contribution to Banach space literature and can be used as a solid introduction to functional analysis ... ." (Stefan Cobzas, Studia Universitatis Babes-Bolyai Mathematica, Vol. XLVII (2), 2002)

"The present book is ... intended as an introduction to linear functional analysis ... . Each chapter concludes with a separate section in which together nearly 700 exercises are listed. Almost all of them, as it seems, are supplemented with hints to their solution. The exercises are an important part of the general text, contain many important results and essentially complement the material in the chapters. ... Altogether, the material presented is simply enormous and could fill with enthusiasm ... ." (J. Synnatzschke, Zeitschrift für Analysis und ihre Anwendungen, Vol. 20 (4), 2001)