Basic real analysis

670 Pages·2005·3.292 MB·English

by  Anthony W. Knapp

Cornerstones
Series Editors
CharlesL.Epstein,UniversityofPennsylvania,Philadelphia
StevenG.Krantz,UniversityofWashington,St.Louis
Advisory Board
AnthonyW.Knapp,StateUniversityofNewYorkatStonyBrook,Emeritus
Anthony W. Knapp
Basic Real Analysis
Along with a companion volume
Advanced Real Analysis
Birkha¨user
Boston • Basel • Berlin
AnthonyW.Knapp
81UpperSheepPastureRoad
EastSetauket,NY11733-1729
U.S.A.
e-mail to: [email protected]
http://www.math.sunysb.edu/˜aknapp/books/basic.html
CoverdesignbyMaryBurgess.
MathematicsSubjectClassicification(2000):28-01,26-01,42-01,54-01,34-01
LibraryofCongressCataloging-in-PublicationData
Knapp,AnthonyW.
Basicrealanalysis:alongwithacompanionvolumeAdvancedrealanalysis/Anthony
W.Knapp
p.cm.–(Cornerstones)
Includesbibliographicalreferencesandindex.
ISBN0-8176-3250-6(alk.paper)
1.Mathematicalanalysis.I.Title.II.Cornerstones(Birkha¨user)
QA300.K562005
515–dc22 2005048070
ISBN-100-8176-3250-6 eISBN0-8176-4441-5 Printedonacid-freepaper.
ISBN-13978-0-8176-3250-2
AdvancedRealAnalysis ISBN0-8176-4382-6
BasicRealAnalysisandAdvancedRealAnalysis(Set) ISBN0-8176-4407-5
(cid:1)c2005AnthonyW.Knapp
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PrintedintheUnitedStatesofAmerica. (MP)
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ToSusan
and
ToMyReal-AnalysisTeachers:
SalomonBochner,WilliamFeller,HillelFurstenberg,
Harish-Chandra,SigurdurHelgason,JohnKemeny,
JohnLamperti,HazletonMirkil,EdwardNelson,
LaurieSnell,EliasStein,RichardWilliamson
CONTENTS
Preface xi
DependenceAmongChapters xiv
GuidefortheReader xv
ListofFigures xviii
Acknowledgments xix
StandardNotation xxi
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 44
8. Summability 53
9. WeierstrassApproximationTheorem 58
10. FourierSeries 61
11. Problems 78
II. METRICSPACES 82
1. DefinitionandExamples 83
2. OpenSetsandClosedSets 91
3. ContinuousFunctions 95
4. SequencesandConvergence 97
5. SubspacesandProducts 102
6. PropertiesofMetricSpaces 105
7. CompactnessandCompleteness 108
8. Connectedness 115
9. BaireCategoryTheorem 117
10. PropertiesofC(S)forCompactMetric S 121
11. Completion 127
12. Problems 130
vii
viii Contents
III. THEORYOFCALCULUSINSEVERALREALVARIABLES 135
1. OperatorNorm 135
2. NonlinearFunctionsandDifferentiation 139
3. Vector-ValuedPartialDerivativesandRiemannIntegrals 146
4. ExponentialofaMatrix 148
5. PartitionsofUnity 151
6. InverseandImplicitFunctionTheorems 152
7. DefinitionandPropertiesofRiemannIntegral 161
8. RiemannIntegrableFunctions 166
9. Fubini’sTheoremfortheRiemannIntegral 169
10. ChangeofVariablesfortheRiemannIntegral 171
11. Problems 179
IV. THEORYOFORDINARYDIFFERENTIALEQUATIONS
ANDSYSTEMS 183
1. QualitativeFeaturesandExamples 183
2. ExistenceandUniqueness 187
3. DependenceonInitialConditionsandParameters 194
4. IntegralCurves 199
5. LinearEquationsandSystems,Wronskian 201
6. HomogeneousEquationswithConstantCoefficients 208
7. HomogeneousSystemswithConstantCoefficients 211
8. SeriesSolutionsintheSecond-OrderLinearCase 218
9. Problems 226
V. LEBESGUEMEASUREANDABSTRACT
MEASURETHEORY 231
1. MeasuresandExamples 231
2. MeasurableFunctions 238
3. LebesgueIntegral 241
4. PropertiesoftheIntegral 245
5. ProofoftheExtensionTheorem 253
6. CompletionofaMeasureSpace 262
7. Fubini’sTheoremfortheLebesgueIntegral 265
8. IntegrationofComplex-ValuedandVector-ValuedFunctions 274
9. L1, L2, L∞,andNormedLinearSpaces 279
10. Problems 289
VI. MEASURETHEORYFOREUCLIDEANSPACE 296
1. LebesgueMeasureandOtherBorelMeasures 297
2. Convolution 306
3. BorelMeasuresonOpenSets 314
4. ComparisonofRiemannandLebesgueIntegrals 318
Contents ix
VI. MEASURETHEORYFOREUCLIDEANSPACE (Continued)
5. ChangeofVariablesfortheLebesgueIntegral 320
6. Hardy–LittlewoodMaximalTheorem 327
7. FourierSeriesandtheRiesz–FischerTheorem 334
8. StieltjesMeasuresontheLine 339
9. FourierSeriesandtheDirichlet–JordanTheorem 346
10. DistributionFunctions 350
11. Problems 352
VII. DIFFERENTIATIONOFLEBESGUEINTEGRALS
ONTHELINE 357
1. DifferentiationofMonotoneFunctions 357
2. AbsoluteContinuity,SingularMeasures,and
LebesgueDecomposition 364
3. Problems 370
VIII.FOURIERTRANSFORMINEUCLIDEANSPACE 373
1. ElementaryProperties 373
2. FourierTransformon L1,InversionFormula 377
3. FourierTransformon L2,PlancherelFormula 381
4. SchwartzSpace 384
5. PoissonSummationFormula 389
6. PoissonIntegralFormula 392
7. HilbertTransform 397
8. Problems 404
IX. Lp SPACES 409
1. InequalitiesandCompleteness 409
2. ConvolutionInvolving Lp 417
3. JordanandHahnDecompositions 418
4. Radon–NikodymTheorem 420
5. ContinuousLinearFunctionalson Lp 424
6. MarcinkiewiczInterpolationTheorem 427
7. Problems 436
X. TOPOLOGICALSPACES 441
1. OpenSetsandConstructionsofTopologies 441
2. PropertiesofTopologicalSpaces 447
3. CompactnessandLocalCompactness 451
4. ProductSpacesandtheTychonoffProductTheorem 458
5. SequencesandNets 463
6. QuotientSpaces 471
x Contents
X. TOPOLOGICALSPACES (Continued)
7. Urysohn’sLemma 474
8. MetrizationintheSeparableCase 476
9. Ascoli–Arzela` andStone–WeierstrassTheorems 477
10. Problems 480
XI. INTEGRATIONONLOCALLYCOMPACTSPACES 485
1. Setting 485
2. RieszRepresentationTheorem 490
3. RegularBorelMeasures 504
4. DualtoSpaceofFiniteSignedMeasures 509
5. Problems 517
XII. HILBERTANDBANACHSPACES 520
1. DefinitionsandExamples 520
2. GeometryofHilbertSpace 526
3. BoundedLinearOperatorsonHilbertSpaces 535
4. Hahn–BanachTheorem 537
5. UniformBoundednessTheorem 543
6. InteriorMappingPrinciple 545
7. Problems 549
APPENDIX 553
A1. SetsandFunctions 553
A2. MeanValueTheoremandSomeConsequences 559
A3. InverseFunctionTheoreminOneVariable 561
A4. ComplexNumbers 563
A5. ClassicalSchwarzInequality 563
A6. EquivalenceRelations 564
A7. LinearTransformations,Matrices,andDeterminants 565
A8. FactorizationandRootsofPolynomials 568
A9. PartialOrderingsandZorn’sLemma 573
A10.Cardinality 577
HintsforSolutionsofProblems 581
SelectedReferences 637
IndexofNotation 639
Index 643