Advanced Real Analysis Along with a companion volume Basic Real Anal
Basic Real Analysis and Advanced Real Analysis (available separately or together as a Set) systematically develop those concepts and tools in real analysis that are vital to every mathematician, whether pure or applied, aspiring or established. These work
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Advisory Board Anthony W. Knapp, State University of New York at Stony Brook, Emeritus
Anthony W. Knapp
Advanced Real Analysis Along with a companion volume Basic Real Analysis
Birkh¨auser Boston • Basel • Berlin
Anthony W. Knapp 81 Upper Sheep Pasture Road East Setauket, NY, 11733-1729 U.S.A. e-mail to: [email protected] http://www.math.sunysb.edu/˜ aknapp/books/advanced.html
Cover design by Mary Burgess. Mathematics Subject Classicification (2000): 46-01, 42-01. 43-01, 35-01, 34-01, 47-01, 58-01, 60A99, 28C10 Library of Congress Cataloging-in-Publication Data Knapp, Anthony W. Advanced real analysis: along with a companion volume Basic real analysis / Anthony W. Knapp p. cm. – (Cornerstones) Includes bibliographical references and index. ISBN 0-8176-4382-6 (alk. paper) 1. Mathematical analysis. I. Title. II. Cornerstones (Birkh¨auser) QA300.K56 2005 515–dc22
2005048070
ISBN-10 0-8176-4382-6 ISBN-13 978-0-8176-4382-9
eISBN 0-8176-4442-3
Basic Real Analysis Basic Real Analysis and Advanced Real Analysis (Set)
Printed on acid-free paper. ISBN 0-8176-3250-6 ISBN 0-8176-4407-5
c 2005 Anthony W. Knapp All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media Inc., 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Printed in the United States of America. 987654321 www.birkhauser.com
SPIN 11372219
(MP)
To Susan and To My Real-Analysis Teachers: Salomon Bochner, William Feller, Hillel Furstenberg, Harish-Chandra, Sigurdur Helgason, John Kemeny, John Lamperti, Hazleton Mirkil, Edward Nelson, Laurie Snell, Elias Stein, Richard Williamson
CONTENTS
List of Figures Preface Dependence Among Chapters Guide for the Reader Notation and Terminology
x xi xiv xv xix
I.
INTRODUCTION TO BOUNDARY-VALUE PROBLEMS 1. Partial Differential Operators 2. Separation of Variables 3. Sturm–Liouville Theory 4. Problems
1 1 3 19 31
II.
COMPACT SELF-ADJOINT OPERATORS 1. Compact Operators 2. Spectral Theorem for Compact Self-Adjoint Operators 3. Hilbert–Schmidt Theorem 4. Unitary Operators 5. Classes of Compact Operators 6. Problems
34 34 36 41 45 46 52
III. TOPICS IN EUCLIDEAN FOURIER ANALYSIS 1. Tempered Distributions 2. Weak Derivatives and Sobolev Spaces 3. Harmonic Functions 4. H p Theory 5. Calder´on–Zygmund Theorem 6. Applications of the Calder´on–Zygmund Theorem 7. Multiple Fourier Series 8. Application to Traces of Integral Operators 9. Problems vii
54 54 60 69 80 83
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