Nonstandard Analysis for the Working Mathematician
Starting with a simple formulation accessible to all mathematicians, this second edition is designed to provide a thorough introduction to nonstandard analysis. Nonstandard analysis is now a well-developed, powerful instrument for solving open problems in
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tandard Analysis for the Working Mathematician Second Edition
Nonstandard Analysis for the Working Mathematician
Peter A. Loeb Manfred P.H. Wolff •
Editors
Nonstandard Analysis for the Working Mathematician Second Edition
123
Editors Peter A. Loeb Department of Mathematics University of Illinois Urbana, IL USA
ISBN 978-94-017-7326-3 DOI 10.1007/978-94-017-7327-0
Manfred P.H. Wolff Mathematical Institute University of Tübingen Tübingen Germany
ISBN 978-94-017-7327-0
(eBook)
Library of Congress Control Number: 2015946066 Primary: 03H05, 03H10, 03H15, 11U10, 12L15, 26E35, 28E05, 46S20, 47S20, 54J05 Secondary: 11B05, 11B13, 11B30, 11B75, 46B08, 46B20, 47A10, 47A58, 47D06, 47H09, 47H10, 54D30, 60G51, 60H07, 60J65, 91A06, 91B99 Springer Dordrecht Heidelberg New York London © Springer Science+Business Media Dordrecht 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Science+Business Media B.V. Dordrecht is part of Springer Science+Business Media (www.springer.com)
Preface
This book is addressed to mathematicians working in analysis, probability, and various applications. The aim is to provide an understandable introduction to the basic theory of nonstandard analysis in Part I, and then illuminate some of the most striking applications. Much of the book, in particular Part I, can be used in a graduate course; problems are posed in those chapters. After Part I, each chapter takes up a different field for the application of nonstandard analysis, beginning with a gentle introduction that even nonexperts can read with profit. The remainder of each chapter is then addressed to experts, showing how to use nonstandard analysis in the search for solutions of open problems and how to obtain rich new structures that produce deep insight into the field under consideration. The applications discussed here are in functional analysis including operator theory, topology applied to compactifications, probability theory including stochastic processes, e
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