剑桥大学――高等数学分析

FurtherAnalysisProf.W.T.GowersLent1997Thesenotesarem[e-mail]:2.8Date:1999/10/2211:33:59Thefollowingpeoplehavemaintainedthesenotes.–datePaulMetcalfeContentsIntroductionv1TopologicalSpaces11.1Introduction...............................11.2BuildingNewSpaces..........................22Compactness52.1Introduction...............................52.2Somecompactsets...........................52.3Consequencesofcompactness.....................72.4Otherformsofcompactness......................73Connectedness93.1Introduction...............................93.2ConnectednessinR...........................103.3Pathconnectedness...........................104Preliminariestocomplexanalysis134.1Paths...................................134.2ComplexIntegration..........................134.3Domains.................................144.4PathIntegrals..............................155Cauchy’stheoremanditsconsequences195.1Cauchy’stheorem............................195.2Homotopy................................205.3ConsequencesofCauchy’sTheorem..................236PowerSeries276.1AnalyticityandHolomorphy......................276.2ClassificationofIsolatedSingularities.................317WindingNumbers357.1IntroductionandDefinition.......................357.2Residues.................................378Cauchy’sTheorem(homologyversion)41iiiivCONTENTSIntroductionThesenotesarebasedonthecourse“FurtherAnalysis”givenbyProf.W.T.Gowers1inCambridgeintheLentTerm1997.ThesetypesetnotesaretotallyunconnectedwithProf.Gowers.Othersetsofnotesareavailablefordifferentcourses.Atthetimeoftypingthesecourseswere:ProbabilityDiscreteMathematicsAnalysisFurtherAnalysisMethodsQuantumMechanicsFluidDynamics1QuadraticMathematicsGeometryDynamicsofD.E.’sFoundationsofQMElectrodynamicsMethodsofMath.PhysFluidDynamics2Waves(etc.)StatisticalPhysicsGeneralRelativityDynamicalSystemsPhysiologicalFluidDynamicsBifurcationsinNonlinearConvectionSlowViscousFlowsTurbulenceandSelf-SimilarityAcousticsNon-NewtonianFluidsSeismicWavesTheymaybedownloadedfrom://@lists.cam.ac.uktogetacopyofthesetsyourequire.1Yes,thatProf.Gowers.vCopyright(c)TheArchimedeans,CambridgeUniversity.Allrightsreserved.Redistributionanduseofthesenotesinelectronicorprintedform,withorwithoutmodification,arepermittedprovidedthatthefollowingconditionsaremet:1.Redistributionsoftheelectronicfilesmustretaintheabovecopyrightnotice,thislistofconditionsandthefollowingdisclaimer.2.Redistributionsinprintedformmustreproducetheabovecopyrightnotice,thislistofconditionsandthefollowingdisclaimer.3.Allmaterialsderivedfromthesenotesmustdisplaythefollowingacknowledge-ment:ThisproductincludesnotesdevelopedbyTheArchimedeans,CambridgeUniversityandtheircontributors.4.NeitherthenameofTheArchimedeansnorthenamesoftheircontributorsmaybeusedtoendorseorpromoteproductsderivedfromthesenotes.5.Neitherthesenotesnoranyderivedproductsmaybesoldonafor-profitbasis,althoughafeemayberequiredforthephysicalactofcopying.6.Youmustcauseanyeditedversionstocarryprominentnoticesstatingthatyoueditedthemandthedateofanychange.THESENOTESAREPROVIDEDBYTHEARCHIMEDEANSANDCONTRIB-UTORS“ASIS”ANDANYEXPRESSORIMPLIEDWARRANTIES,INCLUDING,BUTNOTLIMITEDTO,THEIMPLIEDWARRANTIESOFMERCHANTABIL-ITYANDFITNESSFORAPARTICULARPURPOSEAREDISCLAIMED.INNOEVENTSHALLTHEARCHIMEDEANSORCONTRIBUTORSBELIABLEFORANYDIRECT,INDIRECT,INCIDENTAL,SPECIAL,EXEMPLARY,ORCONSE-QUENTIALDAMAGESHOWEVERCAUSEDANDONANYTHEORYOFLI-ABILITY,WHETHERINCONTRACT,STRICTLIABILITY,ORTORT(INCLUD-INGNEGLIGENCEOROTHERWISE)ARISINGINANYWAYOUTOFTHEUSEOFTHESENOTES,EVENIFADVISEDOFTHEPOSSIBILITYOFSUCHDAM-AGE.Chapter1TopologicalSpaces1.1IntroductionDefinition1.1.AtopologicalspaceisasetXtogetherwithacollectionofsubsetsofXsatisfyingthefollowingaxioms.1.X;2.IfUUn,thenUUn;(thatis,isclosedunderfiniteintersections)3.Anyunionofsetsinisin(orisclosedunderanyunions).Definition1.2.iscalledatopologyonX.Thesetsinarecalledopensets.AsubsetofXisclosedifitscomplementisopen.Examples.1.IfXdisametricspaceandthecollectionofopensets(inametricspacesense)thenXisatopologicalspace.2.IfXisanysetandisthepowersetofX,Xisatopologicalspace.iscalledthediscretetopologyonX.3.IfXisanysetandfXg,Xisatopologicalspace.iscalledtheindiscretetopologyonX.4.IfXisanyinfiniteset,andfYXXnYisfinitegfgthenXisatopologicalspace.iscalledthecofinitetopologyonX.5.IfXisanyuncountableset,andfYXXnYiscountablegfgthenXisatopologicalspace.iscalledthecocountabletopologyonX.Definition1.3.LetAbeasubsetofatopologicalspace.TheclosureofA,denotedA,istheintersectionofallclosedsetscontainingA.NotethatAisclosedandanyclosedsetcontainingAcontainsA.Definition1.4.LetAbeasubsetofatopologicalspace.TheinteriorofA,denotedintAorAistheunionofallopensetsinA.NotethatintAisopenandanyopensetinAisinintA.Definition1.5.TheboundaryAofasetAisAnintA.12CHAPTER1.TOPOLOGICALSPACESDefin