Banach spaces provide a framework for linear and nonlinear functional analysis, operator theory, abstract analysis, probability, optimization and other branches of mathematics. This book introduces the reader to linear functional analysis and to related parts of infinite-dimensional Banach space theory.
- Develops classical theory, including weak topologies, locally convex space, Schauder bases and compact operator theory
- Covers Radon-Nikodým property, finite-dimensional spaces and local theory on tensor products
- Contains sections on uniform homeomorphisms and non-linear theory, Rosenthal's L1 theorem, fixed points, and more
- Includes information about further topics and directions of research and some open problems at the end of each chapter
- Provides numerous exercises for practice
The text is suitable for graduate courses or for independent study. Prerequisites include basic courses in calculus and linear. Researchers in functional analysis will also benefit for this book as it can serve as a reference book.
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All of the authors have previously published with Springer.
Preface.- Basic Concepts in Banach Spaces.- Hahn-Banach and Banach Open Mapping Theorems.- Weak Topologies and Banach Spaces.- Schauder Bases.- Structure of Banach Spaces.- Finite-Dimensional Spaces.- Optimization.- C^1 Smoothness in Separable Spaces.- Superreflexive Spaces.- Higher Order Smoothness.- Dentability and differentiability.- Basics in Nonlinear Geometric Analysis.- Weakly Compactly Generated Spaces.- Topics in Weak Topologies on Banach Spaces.- Compact Operators on Banach Spaces.- Tensor Products.- Appendix.- References.- Symbol Index.- Subject Index.- Author Index.
From the reviews:
"The material touches all the usual introductory topics plus such areas as tensor products, smoothness and other geometric issues, optimization, structure, etc. It is as current as a book this massive and wide-ranging can be. ... it is a critical addition to the library of any college that has functional analysts of any stripe on its campus. ... Summing Up: Essential. Graduate students and researchers/faculty." (D. Robbins, Choice, Vol. 48 (11), July, 2011)
"The book is well-written and is essentially self-contained. All of the standard topics (as well as many other topics) are covered and the authors have accumulated a large collection of exercises on which students can hone their skills. ... an impressive book that should be welcomed by students interested in learning the basic or more advanced topics in the theory of Banach spaces and by researchers in Banach spaces or related fields." (Barry Turett, Zentralblatt MATH, Vol. 1229, 2012)
"It is designed to lead the reader from the basic concepts and principles to several streams of current research in Banach spaces. ... I found the book very readable. It is clearly written and provides accessible references to many techniques that are commonly used in contemporary research in Banach space theory. ... a nice book invaluable both for learning the topic and as a reference. This is definitely a book that anyone interested in Banach space theory (or functional analysis) should have on his/her desk." (Sophocles Mercourakis, Mathematical Reviews, Issue 2012 h)
"This book is a German-style introduction to Banach Spaces. The authors have tried to include everything that might be useful in applications in optimization, PDEs, analysis ... . if you need to know what a dentable Banach space is, you can find out here ... . Most importantly, the book comes with a good set of indices, which should make it a useful reference." (Fernando Q. Gouvea, The Mathematical Association of America, June, 2011)
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