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This page intentionally left blank P1:KDF 0521840724pre CY492/Beals 0521840724 June18,2004 14:4 CharCount=0 ANALYSIS Thisself-containedtext,suitableforadvancedundergraduates,providesanexten- siveintroductiontomathematicalanalysis,fromthefundamentalstomoreadvanced material.Itbeginswiththepropertiesoftherealnumbersandcontinueswitharig- oroustreatmentofsequences,series,metricspaces,andcalculusinonevariable. Further subjects include Lebesgue measure and integration on the line, Fourier analysis, and differential equations. In addition to this core material, the book in- cludesanumberofinterestingapplicationsofthesubjectmattertoareasbothwithin andoutsideofthefieldofmathematics.Theaimthroughoutistostrikeabalance betweenbeingtooaustereortoosketchy,andbeingsodetailedastoobscurethees- sentialideas.Alargenumberofexamplesandnearly500exercisesallowthereader to test understanding and practice mathematical exposition, and they provide a windowintofurthertopics. RichardBealsisJamesE.EnglishProfessorofMathematicsatYaleUniversity.He hasalsoservedasaprofessorattheUniversityofChicagoandasavisitingprofessor attheUniversityofParis,Orsay.Heistheauthorofmorethan100researchpapers andmonographsinpartialdifferentialequations,differentialequations,functional analysis, inverse problems, mathematical physics, mathematical psychology, and mathematicaleconomics. i P1:KDF 0521840724pre CY492/Beals 0521840724 June18,2004 14:4 CharCount=0 ii P1:KDF 0521840724pre CY492/Beals 0521840724 June18,2004 14:4 CharCount=0 ANALYSIS An Introduction RICHARD BEALS YaleUniversity iii    Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge  , UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521840729 © Richard Beals 2004 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2004 - ---- eBook (EBL) - --- eBook (EBL) - ---- hardback - --- hardback - ---- paperback - --- paperback Cambridge University Press has no responsibility for the persistence or accuracy of s for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. P1:KDF 0521840724pre CY492/Beals 0521840724 June18,2004 14:4 CharCount=0 Contents Preface pageix 1 Introduction 1 1A. NotationandMotivation 1 ∗ 1B . TheAlgebraofVariousNumberSystems 5 ∗ 1C . TheLineandCuts 9 1D. Proofs,Generalizations,Abstractions,andPurposes 12 2 TheRealandComplexNumbers 15 2A. TheRealNumbers 15 ∗ 2B . DecimalandOtherExpansions;Countability 21 ∗ 2C . AlgebraicandTranscendentalNumbers 24 2D. TheComplexNumbers 26 3 RealandComplexSequences 30 3A. BoundednessandConvergence 30 3B. UpperandLowerLimits 33 3C. TheCauchyCriterion 35 3D. AlgebraicPropertiesofLimits 37 3E. Subsequences 39 3F. TheExtendedRealsandConvergenceto±∞ 40 3G. SizesofThings:TheLogarithm 42 AdditionalExercisesforChapter3 43 4 Series 45 4A. ConvergenceandAbsoluteConvergence 45 4B. Testsfor(Absolute)Convergence 48 ∗ 4C . ConditionalConvergence 54 ∗ 4D . Euler’sConstantandSummation 57 ∗ 4E . ConditionalConvergence:SummationbyParts 58 AdditionalExercisesforChapter4 59 v P1:KDF 0521840724pre CY492/Beals 0521840724 June18,2004 14:4 CharCount=0 vi Contents 5 PowerSeries 61 5A. PowerSeries,RadiusofConvergence 61 5B. DifferentiationofPowerSeries 63 5C. ProductsandtheExponentialFunction 66 ∗ 5D . Abel’sTheoremandSummation 70 6 MetricSpaces 73 6A. Metrics 73 6B. InteriorPoints,LimitPoints,OpenandClosedSets 75 6C. CoveringsandCompactness 79 6D. Sequences,Completeness,SequentialCompactness 81 ∗ 6E . TheCantorSet 84 7 ContinuousFunctions 86 7A. DefinitionsandGeneralProperties 86 7B. Real-andComplex-ValuedFunctions 90 7C. The Space C(I)91 ∗ 7D . ProofoftheWeierstrassPolynomialApproximationTheorem 95 8 Calculus 99 8A. DifferentialCalculus 99 8B. InverseFunctions 105 8C. IntegralCalculus 107 8D. RiemannSums 112 ∗ 8E . TwoVersionsofTaylor’sTheorem 113 AdditionalExercisesforChapter8 116 9 SomeSpecialFunctions 119 9A. TheComplexExponentialFunctionandRelatedFunctions 119 ∗ 9B . TheFundamentalTheoremofAlgebra 124 ∗ 9C . InfiniteProductsandEuler’sFormulaforSine 125 10 LebesgueMeasureontheLine 131 10A. Introduction 131 10B. OuterMeasure 133 10C. MeasurableSets 136 10D. FundamentalPropertiesofMeasurableSets 139 ∗ 10E . ANonmeasurableSet 142 11 LebesgueIntegrationontheLine 144 11A. MeasurableFunctions 144 ∗ 11B . TwoExamples 148 11C. Integration:SimpleFunctions 149 11D. Integration:MeasurableFunctions 151 11E. ConvergenceTheorems 155 P1:KDF 0521840724pre CY492/Beals 0521840724 June18,2004 14:4 CharCount=0 Contents vii 12 FunctionSpaces 158 12A. NullSetsandtheNotionof“AlmostEverywhere” 158 ∗ 12B . RiemannIntegrationandLebesgueIntegration 159 12C. TheSpace L1 162 12D. TheSpace L2 166 ∗ 12E . DifferentiatingtheIntegral 168 AdditionalExercisesforChapter12 172 13 FourierSeries 173 13A. PeriodicFunctionsandFourierExpansions 173 13B. FourierCoefficientsofIntegrableandSquare-Integrable PeriodicFunctions 176 13C. Dirichlet’sTheorem 180 13D. Feje´r’sTheorem 184 13E. TheWeierstrassApproximationTheorem 187 13F. L2-PeriodicFunctions:TheRiesz-FischerTheorem 189 13G. MoreConvergence 192 ∗ 13H . Convolution 195 ∗ 14 ApplicationsofFourierSeries 197 ∗ 14A . TheGibbsPhenomenon 197 ∗ 14B . AContinuous,NowhereDifferentiableFunction 199 ∗ 14C . TheIsoperimetricInequality 200 ∗ 14D . Weyl’sEquidistributionTheorem 202 ∗ 14E . Strings 203 ∗ 14F . Woodwinds 207 ∗ 14G . SignalsandtheFastFourierTransform 209 ∗ 14H . TheFourierIntegral 211 ∗ 14I . Position,Momentum,andtheUncertaintyPrinciple 215 15 OrdinaryDifferentialEquations 218 15A. Introduction 218 15B. HomogeneousLinearEquations 219 15C. ConstantCoefficientFirst-OrderSystems 223 15D. NonuniquenessandExistence 227 15E. ExistenceandUniqueness 230 15F. LinearEquationsandSystems,Revisited 234 Appendix:TheBanach-TarskiParadox 237 HintsforSomeExercises 241 NotationIndex 255 GeneralIndex 257 P1:KDF 0521840724pre CY492/Beals 0521840724 June18,2004 14:4 CharCount=0 viii