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Title: An Introduction to Banach Space Theory (Graduate Texts in Mathematics, 183) by Robert E. Megginson, Sheldon Axler, F. W. Gehring ISBN: 0-387-98431-3 Publisher: Springer Verlag Pub. Date: September, 1998 Format: Hardcover Volumes: 1 List Price(USD): $74.95

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Average Customer Rating: 5 (2 reviews)

Rating: 5
Summary: Megginson's An Intro to Banach Space Theory
Comment: The editorial and other reviews up to now are very good, and here I want to concentrate on some different aspects. Megginson of the University of Michigan tells the reader that he is preparing the reader for the papers of Lindenstrauss and others. Benyamini and Lindenstrauss have just published a book, via the American Mathematical Society (AMS), Geometric Nonlinear Functional Analysis (2000), which ties in with my work on logic-based probability (LBP) - see abstracts of my papers on the internet at the Institute for Logic of the University of Vienna. I have also described some aspects of functional analysis and semigroups in my other reviews at Amazon.com. Economists and mathematicians are not the only ones who can benefit from this book and the Benyamini-Lindenstrauss book. Anybody interested in speed, acceleration, volume, area, etc., will find the deepest level of analysis being currently explored by functional analysts in Banach Spaces. For the non-specialist, whom I am very interested in addressing, you could think of a Banach space as a space where the objects are speeds, accelerations, volumes, areas, etc., instead of points. One of the astonishing results is that very rare events, events which are contained in other events, boundary events, and lower dimensional events turn out to have critical importance on speeds, accelerations, volumes, areas, etc. The detective story of how this comes about requires knowing the literature on Banach Spaces, and since almost everybody has some contact with speeds, accelerations, volumes, areas, and so on, this means that the non-specialist should hire a consultant or tutor to either translate the material into approximately ordinary English or at least the level of elementary algebra and geometry or to explain the material step by step using a combination of mathematics and English ingenuity. Benyamini and Lindenstrauss, by the way, represent the rapidly oncoming Israeli school of mathematics, and it is interesting that the former British colonies or protectorates (USA, Australia, Canada, New Zealand, Israel)and Great Britain are carrying on much of the research in this area which really started with the speed-acceleration research of Sir Isaac Newton in Great Britain hundreds of years ago. As for the specialist in this area, this (Megginson) is an up to date compilation for graduate students in mathematics, but is also an excellent reference work for Banach Spaces including various integral and derivative spaces and counterexamples and the interesting topics of rotundity, smoothness, weak topology, and nets.

Rating: 5
Summary: the best buy
Comment: Text books of functional analysis usually include Banach space by one or two chapters. But this book is a book of Banach space theory. So you can learn this area more deeply. For mathematical economists, weak and weak star topologies are must-to-learn notions. This book devoted to these topics more than 140 pages. So you can get enough knowledge, I'm convinced. Furthermore, more than 450 exercises will deepen your understanding. Of cource, this book is written for graduate students in mathematics. As the authors say, you can get adequate knowledge if you want to major in Banach space theory.

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