The main theme of this book is the relation between the global structure of Banach spaces and the various types of generalized "coordinate systems" - or "bases" - they possess. This subject is not new and has been investigated since the inception of the study of Banach spaces. In this book, the authors systematically investigate the concepts of Markushevich bases, fundamental systems, total systems and their variants. The material naturally splits into the case of separable Banach spaces, as is treated in the first two chapters, and the nonseparable case, which is covered in the remainder of the book. This book contains new results, and a substantial portion of this material has never before appeared in book form. The book will be of interest to both researchers and graduate students.
Topics covered in this book include:
- Biorthogonal Systems in Separable Banach Spaces - Universality and Szlenk Index - Weak Topologies and Renormings - Biorthogonal Systems in Nonseparable Spaces - Transfinite Sequence Spaces - Applications
Petr Hájek is Professor of Mathematics at the Mathematical Institute of the Academy of Sciences of the Czech Republic. Vicente Montesinos is Professor of Mathematics at the Polytechnic University of Valencia, Spain. Jon Vanderwerff is Professor of Mathematics at La Sierra University, in Riverside, California. Václav Zizler is Professor of Mathematics at the Mathematical Institute of the Academy of Sciences of the Czech Republic.
Topics covered in this book include:
- Biorthogonal Systems in Separable Banach Spaces - Universality and Szlenk Index - Weak Topologies and Renormings - Biorthogonal Systems in Nonseparable Spaces - Transfinite Sequence Spaces - Applications
Petr Hájek is Professor of Mathematics at the Mathematical Institute of the Academy of Sciences of the Czech Republic. Vicente Montesinos is Professor of Mathematics at the Polytechnic University of Valencia, Spain. Jon Vanderwerff is Professor of Mathematics at La Sierra University, in Riverside, California. Václav Zizler is Professor of Mathematics at the Mathematical Institute of the Academy of Sciences of the Czech Republic.
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From the reviews: "This monograph is devoted to the study of the different types of coordinate systems that may exist in infinite-dimensional Banach spaces. ... will certainly become a great reference book for specialists in nonseparable Banach space theory. Its contents are comprehensive and perfectly up to date. Very recent results are included and several proofs are simplified and given with their optimal form. It must be mentioned that this book is also accessible to graduate students and young researchers willing to discover this area." (Gilles Lancien, Mathematical Reviews, Issue 2008 k) "The book under review contains a clear, detailed and self-contained exposition of the modern state-of-the-art in the biorthogonal systems theory. ... one of their goals is to attract young mathematicians to Banach space theory. In my opinion, the book perfectly serves this purpose. ... Every chapter contains an exercises section. Exercises ... are supplied with hints and references to the corresponding literature."(Vladimir Kadets, Zentralblatt MATH, Vol. 1136 (14), 2008)