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Real AnalysisModern Techniques andTheir ApplicationsSecond EditionGerald B. FollandIENCEUp COLLEGE OFLISCBRARYDILIW\NCENTRALI\\Il \1\l\ l \l\l\l I\ I \Il \ l I \ Il ll l \Il l \l\ l\1 1l l\ Il \ Il \ A Wiley-lnterscience PublicationJOHN WILEY & SONS, INC.New York I Chichester I Weinheim I Brisbane I Singapore I Toronto
This book is printed on acid-free paper.Copyright 1999 by John Wiley§& Sons, Inc. All rights reserved.Published simultaneously in Canada.No part of this publication may be reproduced, stored in a retrieval system or transmitted in any formor by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except aspermitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the priorwritten permission of the Publisher, or authorization through payment of the appropriate per-copy fee tothe Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978)& Sons, Inc., 605 Third Avenue, New York , NY 10158-0012, (212) 850-6011 , fax (212)750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department ,John Wiley850-6008. E-Mail: PERMREQ@WILEY.COM.Library of Congress Cataloging-in-Publication Data:Folland, Gerald B.Real analysis: modern techniques and their applicationsFolland. - 2nd ed.p.I Gerald B.em. - (Pure and applied mathematics)"A Wiley-Interscience publication."Includes bibliographical references and index.ISBN 0-471-31716-0 (cloth: alk. paper)1. Mathematical analysis.I. Title.2. Functions of real variables.II. Series: Pure and applied mathematics (John WileySons: Unnumbered)QA300.F67&1999515-dc21Printed in the United States of America10 9 8 7 6 5 4 3 298-37260
To my motherHelen B. Follandand to the memory of my fatherHarold F. Folland
PrefaceThe name "real analysis" is something of an anachronism. Originally applied to thetheory of functions of a real variable, it has come to encompass several subjects ofa more general and abstract nature that underlie much of modem analysis. Thesegeneral theories and their applications are the subject of this book, which is intendedprimarily as a text for a graduate-level analysis course. Chapters 1 through 7 aredevoted to the core material from measure and integration theory, point set topology,and functional analysis that is a part of most graduate curricula in mathematics,together with a few related but less standard items with which I think all analystsshould be acquainted. The last four chapters contain a variety of topics that are meantto introduce some of the other branches of analysis and to illustrate the uses of thepreceding material. I believe these topics are all interesting and important, but theirselection in preference to others is largely a matter of personal predilection.The things one needs to know in order to read this book are as follows:1 . First and foremost, the classical theory of functions of a real variable: limits andcontinuity, differentiation and (Riemann) integration, infinite series, uniformconvergence, and the notion of a metric space.2. The arithmetic of complex numbers and the basic properties of the complexexponential function ex iy ex ( co sy i sin y) . (More advanced resultsfrom complex function theory are used only in the proof of the Riesz-Thorintheorem and in a few exercises and remarks.)3.Some elementary set theory.vii
viiiPREFACE4. A bit of linear algebra - actually, not much beyond the definitions of vectorspaces, linear mappings, and determinants.All of the necessary material in ( 1 ) and (2) can be found in W. Rudin ' s classic Princi ples of Mathematical Analysis (3rd ed., McGraw-Hill, 1976) or its descendants suchas R. S . Strichatrz ' s The Way ofAnalysis (Jones and Bartlett, 1 995) or S. G. Krantz ' sReal Analysis and Foundations (CRC Press, 1 99 1 ). A summary of the relevant factsabout sets and metric spaces is provided here in Chapter 0. The reader should be gin this book by examining §0. 1 and §0.5 to become familiar with my notation andterminology; the rest of Chapter 0 can then be referred to as needed.Each chapter concludes with a section entitled "Notes and References." Thesesections contain miscellaneous remarks, acknowledgments of sources, indicationsof results not discussed in the text, references for further reading, and historicalnotes. The latter are quite sketchy, although references to more detailed sources areprovided; they are intended mainly to give an idea of how the subject grew out of itsclassical origins. I found it entertaining and instructive to read some of the originalpapers, and I hope to encourage others to do the same.A sizable portion of this book is devoted to exercises. They are mostly in theform of assertions to be proved, and they range from trivial to difficult; hints andintermediate steps are provided for the more complicated ones. Every reader shouldperuse them, although only the most ambitious will try to work them all out. Theyserve several purposes: amplification of results and completion of proofs in thetext, discussion of examples and counterexamples, applications of theorems, anddevelopment of further ideas. Instructors will probably wish to do some of theexercises in class; to maximize flexibility and minimize verbosity, I have followedthe principle of "When in doubt, leave it as an exercise," especially with regardto examples. Exercises occur at the end of each section, but they are numberedconsecutively within each chapter. In referring to them, "Exercise n" means the nthexercise in the present chapter unless another section is explicitly mentioned.The topics in the book are arranged so as to allow some flexibility of presentation.For example, Chapters 4 and 5 do not depend on Chapters 1-3 except for a fewexamples and exercises. On the other hand, if one wishes to proceed quickly to LPtheory, one can skip from §3.3 to §§5. 1-2 and thence to Chapter 6. Chapters 1 0and 1 1 are independent of Chapters 8 and 9 except that the ideas i n §8.6 are used inChapter 10.The new features of this edition are as follows: The material on the n-dimensional Lebesgue integral (§§2.6-7) has been rear ranged and expanded.Tychonoff ' s theorem (§4.6) is proved by an elegant argument recently discov ered by Paul Chernoff.The chapter on Fourier analysis has been split into two chapters (8 and 9).The material on Fourier series and integrals (§ §8.3-5) has been rearranged andnow contains the Dirichlet-Jordan theorem on convergence of Fourier series.
PREFACEixThe material on distributions (§§9. 1-2) has been extensively rewritten andexpanded. A section on self-similarity and Hausdorff dimension (§ 1 1 .3) has been added,replacing the outdated calculation of the Hausdorff dimension of Cantor setsin the old § 1 0.2. Innumerable small changes have been made in the hope of improving theexposition.The writer of a text on such a well-developed subject as real analysis must neces sarily be indebted to his predecessors. I kept a large supply of books on hand whilewriting this one; they are too numerous to list here, but most of them can be foundin the bibliography. I am also happy to acknowledge the influence of two of myteachers: the late Lynn Loomis, from whose lectures I first learned this subject, andElias Stein, who has done much to shape my point of view. Finally, I am grateful toa number of people - especially Steven Krantz, Kenneth Ross, and William Faris- whose comments and corrigenda concerning the first edition have helped me toprepare the new one.GERALD B. FOLLANDSeattle, Washington
ContentsPreface0Prologue0.1The Language of Set Theory0.3Cardinality0.2Orderings0.4More about Well Ordered Sets0.60. 7Metric Spaces0.51The Extended Real Number SystemNotes and el Measures on the Real Line1.21.41.6a-algebrasOuter MeasuresNotes and References.Vll1146910131619192124283340xi
xii2CONTENTSIntegration2.12.2Integration of Nonnegative Functions2.4Modes of Convergence2.6Then-dimensional Lebesgue Integral2.3P roduct Measures2.7Integration in Polar Coordinates3.1Signed Measures3.2The Lebesgue-Radon-Nikodym Theorem3. 3Complex Measures3.4Differentiation on Euclidean SpaceFunctions of Bounded VariationNotes and ReferencesPoint Set Topology4.1Topological Spaces4.3Nets4.5Locally Compact Hausdorff Spaces4.7The Stone-Weierstrass Theorem4.24.4Continuous MapsCompact Spaces4.6Two Compactness Theorems4.8Embeddings in Cubes4.95Notes and ReferencesSigned Measures and Differentiation3.53.64Integration of Complex Functions2.52.83Measurable FunctionsNotes and ReferencesElements of Functional Analysis5.1Normed Vector Spaces5.2Linear Functionals5.4Topological Vector Spaces5.3The Baire Category Theorem and its Consequences5.5Hilbert Spaces5.6Notes and 119125128131136138143146151151157161165171179
CONTENTS6LP SpacesBasic Theory of6.3Some Useful ibution Functions and WeakInterpolation of188LPLP SpacesNotes and ReferencesPositive Linear Functionals onCc(X)Regularity and Approximation TheoremsThe Dual ofCo(X)Products of Radon MeasuresNotes and ReferencesElements of Fourier n of Fourier Integrals and Series8.6Fourier Analysis of Measures8.3The Fourier Transform8.5Pointwise Convergence of Fourier Series8.7Applications to Partial Differential Equations8.89The Dual of181Radon Measures7.27.38LP6.1xiiiNotes and ReferencesElements of Distribution 72632702732782819.2DistributionsCompactly Supported, Tempered, and PeriodicDistributions2819.3Sobolev Spaces3019.19.4Notes and References10 Topics in Probability Theory10.1Basic Concepts10.3The Central Limit Theorem10.5The Wiener Process10.2The Law of Large Numbers10.4Construction of Sample Spaces10.6Notes and References291310313313320325328330336
xivCONTENTS11 More Measures and Integrals11.111.211.311.411 .5Topological Groups and Haar MeasureHausdorff MeasureSelf-similarity and Hausdorff DimensionIntegration on ManifoldsNotes and References339339348355361363Bibliography365Index of Notation377Index379
Real Analysis
PrologueThe purpose of this introductory chapter is to establish the notation and terminologythat will be used throughout the book and to present a few diverse results from settheory and analysis that will be needed later. The style here is deliberately terse,since this chapter is intended as a reference rather than a systematic exposition.0.1TH E LANG UAG E OF SET TH EO RYIt is assumed that the reader is familiar with the basic concepts of set theory; thefollowing discussion is meant mainly to fix our terminology.Number Systems .follows:Our notation for the fundamental number systems is asN the set of positive integers (not including zero)Z the set of integers Q the set of rational numberslR the set of real numbers C the set of complex numbersWe shall avoid the use of special symbols from mathematical logic,preferring to remain reasonably close to standard English. We shall, however, usethe abbreviation iff for "if and only if."One point of elementary logic that is often insufficiently appreciated by studentsis the following: If A and B are mathematical assertions and -A, -B are theirLogic.1
2PROLOGUEnegations, the statement "A implies B" is logically equivalent to the contrapositivestatement "-B implies -A." Thus one may prove that A implies B by assuming -Band deducing -A, and we shall frequently do so. This is not the same as reductio adabsurdum, which consists of assuming both A and -B and deriving a contradiction.The words "family" and "collection" will be used synonymously with"set," usually to avoid phrases like "set of sets." The empty set is denoted by 0, andthe family of all subsets of a set X is denoted by P(X):Sets .P(X) { E : E c X } .Here and elsewhere, the inclusion sign c is interpreted in the weak sense; that is, theassertion "E c X" includes the possibility that E X.If is a family of sets, we can form the union and intersection of its members:U E {EE x : xEE for some E E } ,n E {X : X E E for all E E }.EE Usually it is more convenient to consider indexed families of sets:in which case the union and intersection are denoted byIf Ea n Ef3 0 whenever a / {3, the sets Ea are called disjoint. The terms "disjointcollection of sets" and "collection of disjoint sets" are used interchangeably, as are"disjoint union of sets" and "union of disjoint sets."When considering families of sets indexed by N, our usual notation will beand likewise for unions and intersections. In this situation, the notions of limitsuperior and limit inferior are sometimes useful:0000lim sup En n U En ,k l n klim inf En 0000U n En .k l n kThe reader may verify thatlim sup En { x : x E En for infinitely many n} ,lim inf En { x : x E En for all but finitely many n}.
THE LANGUAGE OF SET THEORY3If E and F are sets, we denote their difference by E \ F:E\F { x : x E E and x F } ,and their symmetric difference by E F:E F ( E \ F) u ( F \ E) .When it is clearly understood that all sets in question are subsets of a fixed set X, wedefine the complement Ec of a set E (in X):In this situation we have deMorgan 's laws:( u E, r n E ' aEAnEA( n E, rnEA unEAE .If X and Y are sets, their Cartesian product X x Y is the set of all ordered pairs( x, y) such that x E X and y E Y. A relation from X to Y is a subset of X x Y.(If Y X, we speak of a relation on X.) If R is a relation from X to Y, we shallsometimes write xRy to mean that ( x, y) E R. The most important types of relationsare the following: Equivalence relations. An equivalence relation on X is a relation R on Xsuch thatxRx for all x E X,xRy iff yRx,xRz whenever xRy and yRz for some y .The equivalence class of an element x is {y E X : xRy}. X is the disjointunion of these equivalence classes. Orderings. See §0.2.Mappings. A mapping f : X -t Y is a relation R from X to Y with theproperty that for every x E X there is a unique y E Y such that xRy, in whichcase we write y f ( x) . Mappings are sometimes called maps or functions;we shall generally reserve the latter name for the case when Y is C or somesubset thereof.Z are mappings, we denote by go f their composition:gg o f : X Z, g o f (x) g (f( x) ) .If f : X -t Y and : Y-t -tIf D c X and E c Y, we define the image of D and the inverse image of Eunder a mapping f : X -t Y byf (D) {f ( x) :x E D},f- 1 ( E) { x : f ( x) E E}.
4PROLOGUEIt is easily verified that the map f-1 : P(Y)-t P(X) defined by the second formulac ommutes with union, intersections, and complements:f-1( nUE Ea ) nUE A f-1(Ea),Af-1( nnE Ea) nnE f-1(Ea),AA(The direct image mapping f :P(X)-t P(Y) commutes with unions, but in generalnot with intersections or complements.)Iff :X -t Y is a mapping, X is called the domain of f and f(X) is called therange of f. f is said to be injective if j(x1) j(x 2) only when x 1 x 2, surjectiveif f(X) Y, and bijective if it is both injective and surjective. If f is bijective, ithas an inverse f-1 :Y-t X such that f-1of and fof-1 are the identity mappingson X and Y, respectively. If A c X, we denote by JIA the restriction of f to A:( ! l A ) : A -t Y,(fiA)(x) f(x) for x EA.A sequence in a set X is a mapping from N into X. (We also use the term finitesequence to mean a map from {1, . . . , n } into X where n E N.) If f : N -t X is asequence and g :N -t N satisfies g(n) g(m) whenever n m, the compositionfog is called a subsequence of f . It is common, and often convenient, to be carelessabout distinguishing between sequences and their ranges, which are subsets of Xindexed by N. Thus, if f(n) Xn, we speak of the sequence {xn } ! ; whether wemean a mapping from N to X or a subset of X will be clear from the context.Earlier we defined the Cartesian product of two sets. Similarly one can define theCartesian product of n sets in terms of ordered n-tuples. However, this definitionbecomes awkward for infinite families of sets, so the following approach is usedinstead. If {Xn} nE A is an indexed family of sets, their Cartesian product Ti nE A Xnis the set of all maps f : A-t U nEA Xn such that f(n) E Xn for every n E A. (Itshould be noted, and then promptly forgotten, that when A {1, 2}, the previousdefinition of X1 X x2 is set-theoretically different from the present definition ofTii Xj. Indeed, the latter concept depends on mappings, which are defined in termsof the former one.) If X TinE A Xn and n E A, we define the nth projection orcoordinate map 7rn :X -t Xn by 7rn ( f) f(n). We also frequently write x andXn instead of f and f(n) and call Xn the nth coordinate of x.If the sets Xn are all equal to some fixed set Y, we denote Ti nE A Xn by Y A :Y A the set of all mappings from A to Y.If A { 1 , . . . , n} , Y A is denoted by yn and may be identified with the set of orderedn-tuples of elements of Y.0.2O R D E R INGSA partial ordering on a nonempty set X is a relationproperties:R on X with the following
ORDERINGS if xRy and y Rz, then xRz; if xRy and y Rx, then x xRx for all x. 5y;If R also satisfies if x, y E X, then either xRy or y Rx,then R is called a linear (or total) ordering. For example, if E is any set, then P(E)is partially ordered by inclusion, and IR is linearly ordered by its usual ordering.Taking this last example as a model, we shall usually denote partial orderings by , and we write x y to mean that x y but x / y. We observe that a partialordering on X naturally induces a partial ordering on every nonempty subset of X.Two partially ordered sets X and Y are said to be order isomorphic if there is abijection f : X -t Y such that x 1 x 2 iff j (x 1 ) j (x 2 ).If X is partially ordered by , a maximal (resp. minimal) element of X is anelement x E X such that the only y E X satisfying x y (resp. x y) is x itself.Maximal and minimal elements may or may not exist, and they need not be uniqueunless the ordering is linear. If E c X, an upper (resp. lower) bound for E is anelement x E X such that y x (resp. x y) for all y E E. An upper bound for Eneed not be an element of E, and unless E is linearly ordered, a maximal element ofE need not be an upper bound for E. (The reader should think up some examples.)If X is linearly ordered by and every nonempty subset of X has a (necessarilyunique) minimal element, X is said to be well ordered by , and (in defiance of thelaws of grammar) is called a well ordering on X. For example, N is well orderedby its natural ordering.We now state a fundamental principle of set theory and derive some consequencesof it.0.1 The Hausdorff Maximal Principle. Every partially ordered set has a maximallinearly ordered subset.In more detail, this means that if X is partially ordered by , there is a set E c Xthat is linearly ordered by , such that no subset of X that properly includes E islinearly ordered by . Another version of this principle is the following:0.2 Zorn ' s Lemma. If X is a partially ordered set and every linearly ordered subsetof X has an upper bound, then X has a maximal element.Clearly the Hausdorff maximal principle implies Zorn's lemma: An upper boundfor a maximal linearly ordered subset of X is a maximal element of X. It is also notdifficult to see that Zorn's lemma implies the Hausdorff maximal principle. (ApplyZorn's lemma to the collection of linearly ordered subsets of X, which is partiallyordered by inclusion.)0.3 The Well Ordering Principle. Every nonempty set X can be well ordered.
PROLOGUE6Proof. Let W be the collection of well orderings of subsets of X, and define apartial ordering on W as follows. If 1 and 2 are well orderings on the subsetsE1 and E2 , then 1 precedes 2 in the partial ordering if (i) 2 extends 1 , i.e.,E1 c E2 and 1 and 2 agree on E1 , and (ii) if x E E2 \ E1 then y
For example, Chapters 4 and 5 do not depend on Chapters 1-3 except for a few examples and exercises. On the other hand, if one wishes to proceed quickly to LP theory, one can skip from §3.3 to §§5.1-2 and thence to Chapter 6. Chapters 10 and 11 are independent of Chapters 8 and 9 except that the ideas in §8.6 are used in Chapter 10.
Linacre House, Jordan Hill, Oxford OX2 8DP A division of Reed Educational and Professional Publishing Ltd @_A member of the Reed Elsevier pic group OXFORD BOSTON JOHANNESBURG MELBOURNE NEW DELHI SINGAPORE Translated from the 3rd revised and enlarged edition of Medkani
Real-Time Analysis 1EF77_3e Rohde & Schwarz Implementation of Real -Time Spectrum Analysis 3 1 Real-Time Analysis 1.1 What “Real-Time” Stands for in R&S Real-Time Analyzers The measurement speed available in today's spectrum analyzers is the result of a long
Image restoration techniques are used to make the corrupted image as similar as that of the original image. Figure.3 shows classification of restoration techniques. Basically, restoration techniques are classified into blind restoration techniques and non-blind restoration techniques [15]. Non-blind restoration techniques
asics of real-time PCR 1 1.1 Introduction 2 1.2 Overview of real-time PCR 3 1.3 Overview of real-time PCR components 4 1.4 Real-time PCR analysis technology 6 1.5 Real-time PCR fluorescence detection systems 10 1.6 Melting curve analysis 14 1.7 Passive reference dyes 15 1.8 Contamination prevention 16 1.9 Multiplex real-time PCR 16 1.10 Internal controls and reference genes 18
the model, we hope to overcome the barriers that currently exist in applying real options analysis (ROA) to real estate development, and provide developers with a tool to help them better understand the rewards associated with real options. Real options analysis is an important topic in real estate due to the nature of the development environment.
1.1 Hard Real Time vs. Soft Real Time Hard real time systems and soft real time systems are both used in industry for different tasks [15]. The primary difference between hard real time systems and soft real time systems is that their consequences of missing a deadline dif-fer from each other. For instance, performance (e.g. stability) of a hard real time system such as an avionic control .
techniques were described in chapter 4. It's important to combine two or more diversity techniques to get full advantage of diversity techniques. Some diversity combining techniques were described in chapter 5. The mathematical equations needed for simulation in diversity combining techniques were collected and defined in chapter 6.
AWWA Manual M49 ix Preface The purpose of this manual is to present a recommended method for calculating operating torque, head loss, and cavitation for quarter-turn valves typically used in water works ser-vice. It is a discussion of recommended practice, not an American Water Works Association (AWWA) standard. The text provides guidance on .
