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.
Zusammenfassung
One of the fundamental questions of Banach space theory is whether every Banach space has a basis. A space with a basis gives us the feeling of familiarity and concreteness, and perhaps a chance to attempt the classification of all Banach spaces and other problems.The main goals of this book are to: introduce the reader to some of the basic concepts, results and applications of biorthogonal systems in infinite dimensional geometry of Banach spaces, and in topology and nonlinear analysis in Banach spaces; to do so in a manner accessible to graduate students and researchers who have a foundation in Banach space theory; expose the reader to some current avenues of research in biorthogonal systems in Banach spaces; provide notes and exercises related to the topic, as well as suggesting open problems and possible directions of research.The intended audience will have a basic background in functional analysis. The authors have included numerous exercises, as well as open problems that point to possible directions of research.
Inhalt
Separable Banach Spaces.- Universality and the Szlenk Index.- Review of Weak Topology and Renormings.- Biorthogonal Systems in Nonseparable Spaces.- Markushevich Bases.- Weak Compact Generating.- Transfinite Sequence Spaces.- More Applications.