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electronic publication has now been resolved, and a PDF file, called the “ The core of a first course in complex analysis has been included as Appendix B
International Standard Book Number-13: 978-1-4822-1928-9 (eBook - PDF) Of course, presenting the real number system in this way begs the excellent question real analysis to undergraduates at George Washington University followed the non-starred sections of this text: Chapters 1–7 for the first semester
ambitious reader can get a more general insight either by referring to the book A First Course in Real Analysis or the text Principles of Mathematical Analysis by
Protter Charles B Morrey, Jr A First Course in Real Analysis Second Edition With 143 Illustrations
A First Course in Real Analysis With 19 Illustrations Springer Page 2 Contents Preface vii CHAPTER l Axioms for the Field M of Real Numbers 1 §1 1
2 jan 2016 · algebra, and differential equations to a rigorous real analysis course is a implicit function theorem is motivated by first considering linear
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ambitious reader can get a more general insight either by referring to the book A First Course in Real Analysis or the text Principles of Mathematical Analysis by
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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
quotesdbs_dbs12.pdfusesText_18
[PDF] Real Analysis: A First Course wwwcepuneporg