Biorthogonal Systems in Banach Spaces

ISBN-13: 9781441923950 / Angielski / Miękka / 2011 / 339 str.

Petr Hajek; Vicente Montesinos Santalucia; Jon Vanderwerff

Opis

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."

Recenzje

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)

Spis treści

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.

O autorze

One of the fundamental questions of Banach space theory is whether every Banach space has a basis. A space with a basis gives us a sense of familiarity and concreteness, and perhaps a chance to attempt the classification of all Banach spaces and other problems. 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, in a manner accessible to graduate students and researchers with a foundation in Banach space theory. Numerous exercises are included, and the only prerequisites are a basic background in functional analysis.

Blurb

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.