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1、˖ڍመڙጲhttp://www.paper.edu.cn关于Banach空间中的超弱紧子集和其等价性程立新1，程庆进1，涂昆2，张吉超31厦门大学数学科学学院，厦门市 3610052扬州大学数学科学学院，扬州市 2250003湖北工业大学理学院，武汉市 430068摘要：类比于Banach空间中的弱紧集和超自反空间中子集的性质，本文目的是讨论Banach空间中凸和非凸子集的超弱紧性质。作为结果，本文给出了超弱紧集的两个特征：第一个为Grothendiek型定理；第二个为James型特征。这些结果是通过局部化超积的一些性质以及充分运用Banach空间的一些几何程序得到。关键词：超弱紧
2、集；超积；Banach空间.中图分类号：O177.2OnsuperweakcompactnessofsubsetsanditsequivalencesinBanachspacesCHENGLi-Xin1,CHENGQing-Jin1,TUKun2,ZHANGJi-Chao31SchoolofMathematicalSciences,XiamenUniversity,Xiamen3610052SchoolofMathematicalSciences,YangzhouUniversity,Yanghzou2250003Schoolofscience,HubeiUniversityofTech
3、nology,Wuhan430068Abstract:AnalogoustoweakcompactnessofsubsetsofBanachspacesandtopropertyofsubsetsinsuperreflexivespaces,thepurposeofthispaperistodiscusssuperweakcompactnessofbothconvexandnonconvexsubsetsinBanachspaces.Asaresult,thispapergivestwocharacterizationsofsuperweaklycompactsets:Thefirstone
4、isGrothendiek’stypetheorem;thesecondoneisJames’typecharacterization.ThesearedonebylocalizingsomebasicpropertiesofultrapowersandusingsomegeometricproceduresofBanachspaces.Keywords:superweaklycompactset,ultraproduct,Banachspace.0IntroductionItiswellknownthat,becauseoftheirniceproperties,superreflexi
5、veBanachspaceshaveplayedveryimportantpartingeometryofBanachspacesanditsapplications.RecallthataBanachspaceissaidtobesuperreflexiveprovidedeveryBanachspacefinitelyrepresentableinFoundations:SpecializedResearchFundfortheDoctoralProgramofHigherEducation(20130121110032).AuthorIntroduction:ChengLixin，ma
6、le，professor，majorresearchdirection：FunctionalAnalysis.Email:lx-cheng@xmu.edu.cn.Correspondenceauthor：ChengQingjin,male,associatedprofessor,majorresearchdirection:FunctionalAnalysis.Email:qjcheng@xmu.edu.cn.-1-˖ڍመڙጲhttp://www.paper.edu.cnitisreflexive.In1972,Enflo[1]presentedthefollowingcelebrat
7、edrenormingtheorem:EverysuperreflexiveBanachspacecanberenormedtobeuniformlyconvex.Pisier([2],1975)furthershowedthateverysuperreflexiveBanachspaceisrenormedtohaveaconvexitymodulusofpowertype.Anotherremarkableresultisdueto
