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1、BasicAnalysis–GentlyDoneTopologicalVectorSpacesLectureNotesIvanF.WildeDepartmentofMathematicsKing’sCollege,LondonPrefaceThesenotesformanenhancedversionofthelecturenotesfortheM.Sc.course,BasicAnalysis,presentedintheDepartmentofMathematicsatKing’sColleg
2、e,London.ItisapleasuretothankDr.FrankJellettformanyenjoyablediscussions.Thefollowingbooksmightserveasagoodstartingpointforthosewishingtofindoutmore.(Forexample,Wilansky’sbookincludesasetof33tablesoftheoremsandcounterexamples,aswellas1500problems.)R.V.K
3、adisonandJ.R.Ringrose,FundamentalsoftheTheoryofOperatorAlgebras,Vol.1,AcademicPress,1983,andVol.3,Birkh¨auserBoston,1990J.L.Kelley,GeneralTopology,D.VanNostrand,Princeton,NewJersey,1955;reprinted,Springer-Verlag,NewYork,1975I.J.Maddox,ElementsofFuncti
4、onalAnalysis,CambridgeUniversityPress,2ndedition,1988J.R.Munkres,Topology—AFirstCourse,PrenticeHall,1975M.ReedandB.Simon,MethodsofModernMathematicalPhysics,Vol.1FunctionalAnalysis,AcademicPress,revisedandenlargededition,1980A.P.RobertsonandW.Robertson
5、,TopologicalVectorSpaces,CambridgeUni-versityPress,1964W.Rudin,FunctionalAnalysis,McGraw-Hill,2ndedition,1973G.F.Simmons,IntroductiontoTopologyandModernAnalysis,McGraw-Hill,1963A.E.TaylorandD.C.Lay,IntroductiontoFunctionalAnalysis,Wiley,NewYork,1980A.
6、Wilansky,ModernMethodsinTopologicalVectorSpaces,McGraw-Hill,1978IvanF.WildeiContents1TopologicalSpaces...........................12Nets..................................133ProductSpaces.............................254Separation........................
7、.......325VectorSpaces.............................406TopologicalVectorSpaces.......................537LocallyConvexTopologicalVectorSpaces...............668BanachSpaces.............................849TheDualSpaceofaNormedSpace..................9910Fr
8、´echetSpaces.............................112ii1.TopologicalSpacesThenotionofopensetplaysanimportantrˆoleinthetheoryofmetricspaces.Ourstartingpointistoextractandabstractthebasicpropertiesenjoyedbysuchsets.Thisleadstoawholenewareaofstudy—topolog
