Basic Real Analysis

842 Pages·2016·9.902 MB·English

by  Anthony W. Knapp

BasicRealAnalysis
Digital Second Editions
By Anthony W. Knapp
Basic Algebra
Advanced Algebra
Basic Real Analysis,
with an appendix “ElementaryComplex Analysis”
Advanced Real Analysis
Anthony W. Knapp
BasicRealAnalysis
With an Appendix “Elementary Complex Analysis”
Along with a Companion VolumeAdvancedRealAnalysis
Digital Second Edition, 2016
Published by the Author
East Setauket, New York
AnthonyW.Knapp
81UpperSheepPastureRoad
EastSetauket,N.Y.11733–1729, U.S.A.
Emailto: [email protected]
Homepage:www.math.stonybrook.edu/ aknapp
∼
Title: BasicRealAnalysis,withanappendix“ElementaryComplexAnalysis”
Cover: AninstanceoftheRisingSunLemmainSectionVII.1.
MathematicsSubjectClassification(2010):28–01,26–01,42–01,54–01,34–01,30–01,32–01.
FirstEdition,ISBN-13978-0-8176-3250-2
c2005AnthonyW.Knapp
"
PublishedbyBirkha¨userBoston
DigitalSecondEdition,nottobesold,noISBN
c2016AnthonyW.Knapp
"
Publishedbytheauthor
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iv
ToSusan
and
ToMyChildren,SarahandWilliam,
and
ToMyReal-AnalysisTeachers:
SalomonBochner,WilliamFeller,HillelFurstenberg,
Harish-Chandra,SigurdurHelgason,JohnKemeny,
JohnLamperti,HazletonMirkil,EdwardNelson,
LaurieSnell,EliasStein,RichardWilliamson

CONTENTS
ContentsofAdvancedRealAnalysis xi
DependenceAmongChapters xii
PrefacetotheSecondEdition xiii
PrefacetotheFirstEdition xv
ListofFigures xviii
Acknowledgments xix
GuidefortheReader xxi
StandardNotation xxv
I. THEORYOFCALCULUSINONEREALVARIABLE 1
1. ReviewofRealNumbers,Sequences,Continuity 2
2. InterchangeofLimits 13
3. UniformConvergence 15
4. RiemannIntegral 26
5. Complex-ValuedFunctions 41
6. Taylor’sTheoremwithIntegralRemainder 43
7. PowerSeriesandSpecialFunctions 45
8. Summability 54
9. WeierstrassApproximationTheorem 59
10. FourierSeries 62
11. Problems 78
II. METRICSPACES 83
1. DefinitionandExamples 84
2. OpenSetsandClosedSets 92
3. ContinuousFunctions 96
4. SequencesandConvergence 98
5. SubspacesandProducts 103
6. PropertiesofMetricSpaces 106
7. CompactnessandCompleteness 109
8. Connectedness 116
9. BaireCategoryTheorem 118
10. PropertiesofC(S)forCompactMetric S 122
11. Completion 128
12. Problems 131
vii
viii Contents
III. THEORYOFCALCULUSINSEVERALREALVARIABLES 136
1. OperatorNorm 136
2. NonlinearFunctionsandDifferentiation 140
3. Vector-ValuedPartialDerivativesandRiemannIntegrals 147
4. ExponentialofaMatrix 149
5. PartitionsofUnity 152
6. InverseandImplicitFunctionTheorems 153
7. DefinitionandPropertiesofRiemannIntegral 162
8. RiemannIntegrableFunctions 167
9. Fubini’sTheoremfortheRiemannIntegral 171
10. ChangeofVariablesfortheRiemannIntegral 173
11. ArcLengthandIntegralswithRespecttoArcLength 181
12. LineIntegralsandConservativeVectorFields 194
13. Green’sTheoreminthePlane 203
14. Problems 212
IV. THEORYOFORDINARYDIFFERENTIALEQUATIONS
ANDSYSTEMS 218
1. QualitativeFeaturesandExamples 218
2. ExistenceandUniqueness 222
3. DependenceonInitialConditionsandParameters 229
4. IntegralCurves 234
5. LinearEquationsandSystems,Wronskian 236
6. HomogeneousEquationswithConstantCoefficients 243
7. HomogeneousSystemswithConstantCoefficients 246
8. SeriesSolutionsintheSecond-OrderLinearCase 253
9. Problems 261
V. LEBESGUEMEASUREANDABSTRACT
MEASURETHEORY 267
1. MeasuresandExamples 267
2. MeasurableFunctions 274
3. LebesgueIntegral 277
4. PropertiesoftheIntegral 281
5. ProofoftheExtensionTheorem 289
6. CompletionofaMeasureSpace 298
7. Fubini’sTheoremfortheLebesgueIntegral 301
8. IntegrationofComplex-ValuedandVector-ValuedFunctions 310
9. L1, L2, L ,andNormedLinearSpaces 315
∞
10. ArcLengthandLebesgueIntegration 325
11. Problems 327