Real Analysis
The focus of this modern graduate text in real analysis is to prepare the potential researcher to a rigorous "way of thinking" in applied mathematics and partial differential equations. The book will provide excellent foundations and serve as a solid buil
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Edited by Herbert Amann, University of Zurich Steven G. Krantz, Washington University Shrawan Kumar, University of North Carolina at Chapel Hill
Emmanuele DiBenedetto
Real Analysis
Springer Science+Business Media, LLC
Emmanuele DiBenedetto Department of Mathematics Vanderbilt Vniversity Nashvilie, TN 37240
V.S.A.
Library of Congress Cataloging-in-Publication Data
OiBenedetto, Emmanuele. Real Analysis / Emmanuele OiBenedetto. p. cm. - (Birkhăuser advanced texts) Includes bibliographical references and index. ISBN 978-1-4612-6620-4 ISBN 978-1-4612-0117-5 (eBook) DOI 10.1007/978-1-4612-0117-5 1. Mathematical analysis. I. Title. II. Series.
QA300.046 2001 515-dc21
2001052752 CIP
AMS Classification Cades: 03E04, 03EIO, 03E20, 03E25, 26A03, 26A09, 26A12. 26A15, 26A16, 26A21. 26A27, 26A30, 26A42, 26A45, 26A46, 26A48, 26A51, 26805, 26815, 26820, 26B25, 26830.26835, 26B4O. 26EI0. 28A05,28AIO, 28A12, 28A15, 28A20. 28A25,28A33. 28A35, 28A50. 28A75,28A78,31B05, 31B 10, 35C15, 35E05. 4OA05,40AlO, 41A 10,42825,42835, 46A03, 46A22, 46A30, 46A32, 46B03, 46B07, 46B 10, 46825, 46B45, 46C05, 46C 15,46E05. 46EIO, 46E15, 46E35. 46F05, 46FIO, 54A05, 54AIO, 54A20, 54A25, 54B05, 54BIO, 54B15, 54C05. 54C30, 54D05, 54010. 54030, 54045, 54060, 54065, 54E35, 54E40, 54E45, 54E50, 54E52 Printed on acid-ti-ee paper © 2002 Springer Science+Business Media New York Originally published by Birkhlluser 80ston in 2002 Softcover reprint ofthe hardcover 1st edition 2002 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, 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. ISBN 978-1-4612-6620-4
SPIN 10798100
Reformatted from the author's files by John Spiegelman, Philadelphia, PA. 987 6 5 4 3 2 I
Contents
Preface Acknowledgments
xv xxiii
Preliminaries Countable sets 2 The Cantor set 3 Cardinality.. 3.1 Some examples 4 Cardinality of some infinite Cartesian products 5 Orderings, the maximal principle, and the axiom of choice 6 Well-ordering....... 6.1 The first uncountable Problems and Complements
2 4 5 6 8 9 11 II
I
17
2 3 4 5
Topologies and Metric Spaces Topological spaces . . . . . . 1.1 Hausdorff and normal spaces . Urysohn's lemma . . . . . . . . . . The Tietze extension theorem Bases, axioms of countability, and product topologies. 4.1 Product topologies . . . . . . . . . . . . Compact topological spaces . . . . . . . . . . 5.1 Sequentially compact topological spaces.
1
17 19 19 21 22 24 25
26
vi
Contents 6 7 8 9
Compact subsets of ]RN . . . . .. . .
