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Orthonormal Systems and Banach Space Geometry describes the interplay between orthonormal expansions and Banach space geometry. Using harmonic analysis as a starting platform, classical inequalities and special functions are used to study orthonormal systems leading to an understanding of the advantages of systems consisting of characters on compact Abelian groups. Probabilistic concepts such as random variables and martingales are employed and Ramsey's theorem is used to study the theory of super-reflexivity. The text yields a detailed insight into concepts including type and co-type of Banach spaces, B-convexity, super-reflexivity, the vector-valued Fourier transform, the vector-valued Hilbert transform and the unconditionality property for martingale differences (UMD). A long list of unsolved problems is included as a starting point for research. This book should be accessible to graduate students and researchers with some basic knowledge of Banach space theory, real analysis, probability and algebra.
This book is based on the pioneering work of (in chronological order) R. C. James, S. Kwapień, B. Maurey, G. Pisier, D. L. Burkholder and J. Bourgain.
We have done our best to unify and simplify the material. All participants of the Jenaer Seminar ‘Operatorenideale’ contributed their ideas and their patience. Above all, we are indebted to A. Hinrichs who made several significant improvements. From S. Geiss we learnt many results and techniques related to the theory of martingales. Particular gratitude goes to H. Jarchow (Zürich) for various helpful remarks.
For many years, our research on this subject was supported by the Deutsche Forschungsgemeinschaft, contracts Ko 962/3–1 and Pi 322/1–1.
Finally, we thank CAMBRIDGE UNIVERSITY PRESS for their excellent collaboration in publishing this book. Jena, October 1997 ALBRECHT PIETSCH JÖRG WENZEL