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1、ApplicationsofApproachTheoryinFunctionalAnalysisR.Lowen,M.SioenUniversiteitAntwerpen,RUCADepartementWiskundeenInformaticaGroenenborgerlaan1712020Antwerpen,Belgiume-mail:lowen@ruca.ua.ac.be,sioen@ruca.ua.ac.beAbstractStartingfromanormedrealvectorspace(X;jjjj),weintroduceauniformapproach
2、structureonXthetopologicaland1p-metriccore-0
ectionofwhicharetheweaktopologyandthemetricd.We(X;X)jjjjinvestigatethepropertiesofthisintroducedapproachstructure,obtainingquantitativeresultswhichimplytheirclassicalqualitativecounterparts.AMSSubjectClassi cation:46B10,46M99,54B30,54E99Keyw
3、ordsandphrases:Normedspace,weaktopology,weak*to-pology,uniformapproachspace,re
exivity,measureofcompactness,completeless,completion1IntroductionInthestudyofnormedlinearspacesandtheirduality,theso-calledweaktopol-ogyplaysanimportantrole.Itisoutofnecessity,unfortunately,thatweloosethecanon
4、icalnumericalinformationwedisposeofinthegivennormedspacewhenconsideringthisweakstructure.Thereasonforthisisthatitisformedoutoftheoriginalnormedobjectbytakinganinitiallifting,whichofcoursecanbeconstructedonthetopological,butfailstoexistonthenormedormetriclevel.Thesolutionweinvestigateinth
5、ispaperissteppingoutsidethetopologicalsettingintothebroadernumericalframeworkofapproachspacesasintroducedin(R.Lowen[10]),inwhichtheseinitialliftingscanbeperformedinawaycompatiblewiththewell-studiedtopologicalobjects.Itisouraimtopresentastudyofthebasicpropertiesoftheintroducedstructureand
6、toshowthatmanyclassicalresultsnowcanbeobtainedassimplecorollaries.AspirantvanhetBelgischNationaalFondsvoorWetenschappelijkOnderzoek251ApplicationsofApproachTheoryinFunctionalAnalysis2522PreliminariesIn(R.Lowen[10])"approachspaces"wereintroducedtoanswerthequestionwhatnumericalinformation
7、canbepreserved,whilemaintainingcompatibiltywiththeunderlyingtopologies,whenconstructingproducts,quotientsandco-productsofmetricspaces,sincethemetricstructureisextremelyunstablewithrespecttothementionedconstructions.Foranyinformation,notations,de ni-tionsandtheoremsconcern
