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Biorthogonal Systems in Banach Spaces / Edition 1

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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.

About the Author:
Petr Hajek is Professor of Mathematics at the Mathematical Institute of the Academy of Sciences of the Czech Republic

About the Author:
Vicente Montesinos Santalucia is Professor of Mathematics at the Universidad Politecnica de Valencia, Spain

About the Author:
Jon Vanderwerff is Professor of Mathematics at La Sierra University, in Riverside, California

About the Author:
Vaclav Zizler is Professor of Mathematics at the Mathematical Institute of the Academy of Sciences of the Czech Republic

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Editorial Reviews

From the Publisher

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)

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Product Details

  • ISBN-13: 9780387689142
  • Publisher: Springer New York
  • Publication date: 9/28/2007
  • Series: CMS Books in Mathematics Series
  • Edition description: 2008
  • Edition number: 1
  • Pages: 339
  • Product dimensions: 6.20 (w) x 9.30 (h) x 0.90 (d)

Table of Contents

Preface     VII
Standard Definitions, Notation, and Conventions     XVII
Separable Banach Spaces     1
Basics     2
Auerbach Bases     5
Existence of M-bases in Separable Spaces     8
Bounded Minimal Systems     9
Strong M-bases     21
Extensions of M-bases ..     29
[omega]-independence     38
Exercises     42
Universality and the Szlenk Index     45
Trees in Polish Spaces     46
Universality for Separable Spaces     49
Universality of M-bases     57
Szlenk Index     62
Szlenk Index Applications to Universality     70
Classification of C[0, [alpha]] Spaces     73
Szlenk Index and Renormings     77
Exercises     82
Review of Weak Topology and Renormings     87
The Dual Mackey Topology     88
Sequential Agreement of Topologies in X*     92
Weak Compactness in ca([Sigma]) and L[subscript 1]([lamda])     95
Decompositions of Nonseparable Banach Spaces     102
Some Renorming Techniques     107
A Quantitative Version of Krein's Theorem     119
Exercises     125
Biorthogonal Systems in Nonseparable Spaces     131
Long Schauder Bases     132
Fundamental Biorthogonal Systems     137
Uncountable Biorthogonal Systems in ZFC     143
Nonexistence of Uncountable Biorthogonal Systems     148
Fundamental Systems under Martin's Axiom     152
Uncountable Auerbach Bases     158
Exercises     161
Markushevich Bases     165
Existence of Markushevich Bases     165
M-bases with Additional Properties     170
[Sigma]-subsets of Compact Spaces     176
WLD Banach Spaces and Plichko Spaces     179
C(K) Spaces that Are WLD     187
Extending M-bases from Subspaces     191
Quasicomplements     197
Exercises     203
Weak Compact Generating     207
Reflexive and WCG Asplund Spaces     207
Reflexive Generated and Vasak Spaces     212
Hilbert Generated Spaces     225
Strongly Reflexive and Superreflexive Generated Spaces     233
Exercises     239
Transfinite Sequence Spaces     241
Disjointization of Measures and Applications     241
Banach Spaces Containing l[subscript 1]([Gcy])     252
Long Unconditional Bases     259
Long Symmetric Bases     266
Exercises     270
More Applications     273
Biorthogonal Systems and Support Sets     273
Kunen-Shelah Properties in Banach Spaces     276
Norm-Attaining Operators     284
Mazur Intersection Properties     289
Banach Spaces with only Trivial Isometries     297
Exercises     300
References     303
Symbol Index     323
Subject Index     327
Author Index     335
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