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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 1.1.Approximateidentity60 1.2.Fourierseriesofsawtoothfunction66 1.3.Dirichletkernel70 3.2.Cycloid193 3.3.Apiecewise C1 curvethatretracespartofitself200
3.4.Green'sTheoremforanannulus206 surfacearea210 equations220
4.2.Integralcurveofavectorfield234 4.3.GraphofBesselfunction J0 (t)260
6.1.ConstructionofaCantorfunction F381
7.1.RisingSunLemma396 ConvexityTheorem468 B.1.Firstbisectionoftherectangle RinGoursat'sLemma642 B.4.Computationof z)∅1 d?overastandardcircle when zisnotthecenter647
B.5.Constructionofacycle ?andapoint awith n(?,a)=1684
B.6.Twopolygonalpaths ∩1 and ∩2 from z0 to zintheregion U686 (b)asthesumofthreecycles696 xviii
ACKNOWLEDGMENTS UniversityforSpring1963. TrigonometricSeries. CalculusonManifolds. Analysis. Series. xix xxAcknowledgments
Nelson'sProbability. Analysis,Riesz-Nagy'sFunctionalAnalysis. quotesdbs_dbs17.pdfusesText_23
[PDF] Real Analysis A Long Form Mathematics Textbook