Biorthogonal Systems in Banach Spaces

The main theme of this book is the relation between the global structureof Banach spaces and the various types of generalized "coordinatesystems" - or "bases" - they possess. This subject is not new and hasbeen investigated since the inception of the study of Banach spaces. Inthis book, the authors systematically investigate the concepts ofMarkushevich bases, fundamental systems, total systems and theirvariants. The material naturally splits into the case of separableBanach spaces, as is treated in the first two chapters, and thenonseparable case, which is covered in the remainder of the book. Thisbook contains new results, and a substantial portion of this materialhas never before appeared in book form. The book will be of interest toboth 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 ofthe Academy of Sciences of the Czech Republic. Vicente Montesinos isProfessor of Mathematics at the Polytechnic University ofValencia, Spain. Jon Vanderwerff is Professor of Mathematics at LaSierra University, in Riverside, California. Václav Zizler is Professorof Mathematics at the Mathematical Institute of the Academy of Sciencesof the Czech Republic.This book introduces 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. It achieves this in a manner accessible to graduate students and researchers who have a foundation in Banach space theory. The authors have included numerous exercises, as well as open problems that point to possible directions of research.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.