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This book is based on a course in real analysis offered to advanced undergraduates and first-year graduate students at Bowling Green State University

:

Basic Real Analysis

DigitalSecondEditions

ByAnthonyW.Knapp

BasicAlgebra

AdvancedAlgebra

BasicRealAnalysis,

AdvancedRealAnalysis

Anthony W. Knapp

Basic Real Analysis

With an Appendix ÒElementary Complex AnalysisÓ

Along with a Companion Volume

Advanced Real Analysis

Digital Second Edition, 2016

Published by the Author

East Setauket, New York

AnthonyW.Knapp

81UpperSheepPastureRoad

EastSetauket,N.Y.11733-1729,U.S.A.

Emailto:aknapp@math.stonybrook.edu

Homepage:www.math.stonybrook.edu/?aknapp

FirstEdition,ISBN-13978-0-8176-3250-2

cδ2005AnthonyW.Knapp

PublishedbyBirkh¨auserBoston

DigitalSecondEdition,nottobesold,noISBN

cδ2016AnthonyW.Knapp

Publishedbytheauthor

writtenpermissionfromtheauthor.

MediaInc.

iv

ToSusan

and

ToMyChildren,SarahandWilliam,

and

ToMyReal-AnalysisTeachers:

LaurieSnell,EliasStein,RichardWilliamson

CONTENTS

ContentsofAdvancedRealAnalysisxi

DependenceAmongChaptersxii

PrefacetotheSecondEditionxiii

PrefacetotheFirstEditionxv

ListofFiguresxviii

Acknowledgmentsxix

GuidefortheReaderxxi

StandardNotationxxv

I.THEORYOFCALCULUSINONEREALVARIABLE1

2.InterchangeofLimits13

3.UniformConvergence15

4.RiemannIntegral26

5.Complex-ValuedFunctions41

6.Taylor'sTheoremwithIntegralRemainder43

7.PowerSeriesandSpecialFunctions45

8.Summability54

9.WeierstrassApproximationTheorem59

10.FourierSeries62

11.Problems78

II.METRICSPACES83

1.DefinitionandExamples84

2.OpenSetsandClosedSets92

3.ContinuousFunctions96

4.SequencesandConvergence98

5.SubspacesandProducts103

6.PropertiesofMetricSpaces106

7.CompactnessandCompleteness109

8.Connectedness116

9.BaireCategoryTheorem118

10.Propertiesof

C(S)forCompactMetric

S122

11.Completion128

12.Problems131

vii viiiContents

1.OperatorNorm136

4.ExponentialofaMatrix149

5.PartitionsofUnity152

6.InverseandImplicitFunctionTheorems153

8.RiemannIntegrableFunctions167

13.Green'sTheoreminthePlane203

14.Problems212

IV.THEORYOFORDINARYDIFFERENTIALEQUATIONS

ANDSYSTEMS218

1.QualitativeFeaturesandExamples218

2.ExistenceandUniqueness222

4.IntegralCurves234

5.LinearEquationsandSystems,Wronskian236

9.Problems261

V.LEBESGUEMEASUREANDABSTRACT

MEASURETHEORY267

1.MeasuresandExamples267

2.MeasurableFunctions274

3.LebesgueIntegral277

4.PropertiesoftheIntegral281

5.ProofoftheExtensionTheorem289

6.CompletionofaMeasureSpace298

9. L1 L2

L∅

,andNormedLinearSpaces315

10.ArcLengthandLebesgueIntegration325

11.Problems327

Contentsix

VI.MEASURETHEORYFOREUCLIDEANSPACE334

2.Convolution344

3.BorelMeasuresonOpenSets352

6.Hardy-LittlewoodMaximalTheorem365

8.StieltjesMeasuresontheLine377

10.DistributionFunctions388

11.Problems390

VII.DIFFERENTIATIONOFLEBESGUEINTEGRALS

ONTHELINE395

1.DifferentiationofMonotoneFunctions395

LebesgueDecomposition402

3.Problems408

VIII.FOURIERTRANSFORMINEUCLIDEANSPACE411

1.ElementaryProperties411

2.FourierTransformon

L1 ,InversionFormula415

3.FourierTransformon

L2 ,PlancherelFormula419

4.SchwartzSpace422

5.PoissonSummationFormula427

6.PoissonIntegralFormula430

7.HilbertTransform435

8.Problems442

IX.

LpSPACES448

1.InequalitiesandCompleteness448

2.ConvolutionInvolving

Lp456

3.JordanandHahnDecompositions458

4.Radon-NikodymTheorem459

5.ContinuousLinearFunctionalson

Lp463

6.Riesz-ThorinConvexityTheorem466

7.MarcinkiewiczInterpolationTheorem476

8.Problems484

xContents

X.TOPOLOGICALSPACES490

2.PropertiesofTopologicalSpaces496

3.CompactnessandLocalCompactness500

5.SequencesandNets512

6.QuotientSpaces520

7.Urysohn'sLemma523

8.MetrizationintheSeparableCase525

10.Problems529

XI.INTEGRATIONONLOCALLYCOMPACTSPACES534

1.Setting534

2.RieszRepresentationTheorem539

3.RegularBorelMeasures553

4.DualtoSpaceofFiniteSignedMeasures558

5.Problems566

XII.HILBERTANDBANACHSPACES570

1.DefinitionsandExamples570

2.GeometryofHilbertSpace576

4.Hahn-BanachTheorem587

5.UniformBoundednessTheorem593

6.InteriorMappingPrinciple595

7.Problems599

APPENDIXA.BACKGROUNDTOPICS603

A1.SetsandFunctions603

A4.ComplexNumbers613

A5.ClassicalSchwarzInequality614

A6.EquivalenceRelations614

A8.FactorizationandRootsofPolynomials618

A9.PartialOrderingsandZorn'sLemma623

A10.Cardinality627

Contentsxi

APPENDIXB.ELEMENTARYCOMPLEXANALYSIS631

B2.ComplexLineIntegrals636

B4.CauchyIntegralFormula648

B5.Taylor'sTheorem654

B6.LocalPropertiesofAnalyticFunctions656

B7.LogarithmsandWindingNumbers660

B8.OperationsonTaylorSeries665

B9.ArgumentPrinciple669

B10.ResidueTheorem673

B11.EvaluationofDefiniteIntegrals675

B13.GlobalTheoremsinGeneralRegions694

B14.LaurentSeries696

B16.Problems704

HintsforSolutionsofProblems715

SelectedReferences793

IndexofNotation795

Index799

CONTENTSOFADVANCEDREALANALYSIS

I.IntroductiontoBoundary-ValueProblems

II.CompactSelf-AdjointOperators

III.TopicsinEuclideanFourierAnalysis

IV.TopicsinFunctionalAnalysis

V.Distributions

VI.CompactandLocallyCompactGroups

VIII.AnalysisonManifolds

IX.FoundationsofProbability

X.IntroductiontoWavelets

DEPENDENCEAMONGCHAPTERS

I,II,IIIinorder

VIV VI

VIIIVIIX

IX

IX.6XI

XII xii

PREFACETOTHESECONDEDITION

aboutincludingthesetopics. includedwithoutrenumbering. elsewhere. beenincluded. xiii xivPrefacetotheSecondEdition

Theorem,afundamentalresultaboutL

p spacesthattakesadvantageofele- mentarycomplexanalysis. earlierproofhavinghadagap. necessary. madeanumberofsuggestions

Mathematica.

NAPP

February2016

PREFACETOTHEFIRSTEDITION

tolearnbyself-study. narydifferentialequations. thebasicsofHilbertandBanachspaces. •ThesubjectsofFourierseriesand harmonicfunctionsareusedasrecurring xv xviPrefacetotheFirstEdition many. quotientishelpfulaswell. variables,leavingoutonlythemorefamiliarpartsnearthebeginning - suchas integralfromChapterViftimepermits.

Typicallythismaterialis

p spacesandintegrationon

PrefacetotheFirstEditionxvii

section"GuidetotheReader"onpagesxv-xvii. butnoattempthasbeen thoseoriginallecturenotes. thefiguresweredrawnwithMathematica. ofknowncorrectionsonmyownWebpage. A.W.K NAPP

May2005

LISTOFFIGURES

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