关于Banach空间中的超弱紧子集和其等价性.pdf 13页

'˖ڍመ੾᝶஠ڙጲhttp://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：超弱紧集；超积；Banach空间.中图分类号：O177.2OnsuperweakcompactnessofsubsetsanditsequivalencesinBanachspacesCHENGLi-Xin1,CHENGQing-Jin1,TUKun2,ZHANGJi-Chao31SchoolofMathematicalSciences,XiamenUniversity,Xiamen3610052SchoolofMathematicalSciences,YangzhouUniversity,Yanghzou2250003Schoolofscience,HubeiUniversityofTechnology,Wuhan430068Abstract:AnalogoustoweakcompactnessofsubsetsofBanachspacesandtopropertyofsubsetsinsuperreflexivespaces,thepurposeofthispaperistodiscusssuperweakcompactnessofbothconvexandnonconvexsubsetsinBanachspaces.Asaresult,thispapergivestwocharacterizationsofsuperweaklycompactsets:ThefirstoneisGrothendiek’stypetheorem;thesecondoneisJames’typecharacterization.ThesearedonebylocalizingsomebasicpropertiesofultrapowersandusingsomegeometricproceduresofBanachspaces.Keywords:superweaklycompactset,ultraproduct,Banachspace.0IntroductionItiswellknownthat,becauseoftheirniceproperties,superreflexiveBanachspaceshaveplayedveryimportantpartingeometryofBanachspacesanditsapplications.RecallthataBanachspaceissaidtobesuperreflexiveprovidedeveryBanachspacefinitelyrepresentableinFoundations:SpecializedResearchFundfortheDoctoralProgramofHigherEducation(20130121110032).AuthorIntroduction:ChengLixin，male，professor，majorresearchdirection：FunctionalAnalysis.Email:lx-cheng@xmu.edu.cn.Correspondenceauthor：ChengQingjin,male,associatedprofessor,majorresearchdirection:FunctionalAnalysis.Email:qjcheng@xmu.edu.cn.-1- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnitisreflexive.In1972,Enflo[1]presentedthefollowingcelebratedrenormingtheorem:EverysuperreflexiveBanachspacecanberenormedtobeuniformlyconvex.Pisier([2],1975)furthershowedthateverysuperreflexiveBanachspaceisrenormedtohaveaconvexitymodulusofpowertype.AnotherremarkableresultisduetoJames[3]:ABanachspaceissuperreflexiveifandonlyifitdoesnotadmitfinitetreeproperty,i.e.foreveryε>0theclosedunitballdoesnotadmit(n,ε)-treeforallsufficientlylargen∈N.AslocalizationsofsuperreflexivityofBanachspaces,analogoustotheweakcompactnessofsubsetsinBanachspaces,anumberofnotionshavebeenintroducedandstudiedsincemiddle70’softhelastcentury.Beauzamy[4,1976]introducedthenotionofuniformlyconvexifyingoperatorsT∈B(X,Y)forBanachspacesXandY:T(BX)doesnotadmitfinitetreeproperty.HeshowedthatTisuniformlyconvexifyingifandonlyifXadmitsanequivalentnorm∥·∥suchthatforeveryε>0thereisδ>0sothat∥T(u−v)∥<εwheneveru,v∈Xsatisfy∥u∥=∥v∥=1and∥u+v∥>2−δ.Sincethen,manydifferentandgeneralizednotionsappeared(see,forinstance,[5,6,7,8]).Raja[9,2008]introducedthenotionoffinitelydentablemap,whichcanbeunderstoodasageneralizationofBeauzamy’suniformlyconvexifyingoperatortononlinearcase.HeshowedthatanoperatorT∈B(X,Y)isuniformlyconvexifyingifandonlyifT(BX)isfinitelydentableinY;AboundedclosedconvexsetC⊂YisfinitelydentableifandonlyifC⊂T(BX)forsomeBanachspaceXanduniformlyconvexifyingoperatorT∈B(X,Y).Moreover,hegaveCepedello’stheorem[10](see,also,[11,p.94])alocalization:AboundedclosedconvexsetCisfinitelydentableifandonlyifithasuniform∆-convexapproximatingproperty,i.e.itsatisfiesthateveryLipschitzfunctiononitcanbeuniformlyapproximatedbyasequenceofdifferencesofconvexLipschitzfunctions.Wealsocallafinitelydentablesetadmittingfiniteindexproperty.Forstudyofrenormingpropertiesofsmoothness,Fabian,MontesinosandZizler[12,2009]introducedthenotionoffiniteε-dualindexofboundednonemptysubsets.Wealsocallaboundedsubsetadmittingfiniteε-dualindexpossessingfinitedualindexproperty.Amongmanyotherthings,theyshowedaboundedsubsetMofaBanachspaceXhasfiniteε-dualindexifandonlyifXpossessesanequivalentM-uniformlyGateauxsmoothnorm.In2010,throughalocalizedsettingofthenotionoffiniterepresentability,Cheng,Cheng,WangandZhang[13]introducedthenotionofsuperweakcompactnessforclosedboundedconvexsets.AnalogoustoJames’characterizationsofsuperreflexiveBanachspaces[3],theyshowedthataboundedclosedconvexsubsetCisnotsuperweaklycompactifandonlyifCadmitsfinitetreeproperty,whichisequivalenttothefollowinguniformfiniteseparatingproperty:Thereexistsθ>0satisfyingthatforalln∈N,thereisasequence{x}n⊂Csuchii=1thatdist(co{x1,···,xk},co{xk+1,···,xn})≥θ,forall1≤k0andforeverysimplexSBwithverticesinBthereexistasimplexSAwithverticesinA,andanaffinemappingT:aff(SB)→aff(SA)suchthatT(SB)=SAandsuchthat(1−ε)∥x−y∥≤∥Tx−Ty∥≤(1+ε)∥x−y∥,∀x,y∈aff(SB).(2)ii)BissaidtobelinearlyfinitelyrepresentableinAifforeveryε>0andforevery-3- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnsimplexSBwithverticesinBthereexistasimplexSAwithverticesinA,andalinearmappingT:span(SB)→span(SA)suchthatT(SB)=SAandsuchthat(2)holdsonspan(SB).iii)BiscrudelyfinitelyrepresentableinAifthereisaconstantβ≥1sothatforeverysimplexSBwithverticesinBthereexistasimplexSAwithverticesinA,andanaffinemappingT:aff(SB)→aff(SA)suchthatT(SB)=SAandsuchthat−1β∥x−y∥≤∥Tx−Ty∥≤β∥x−y∥,∀x,y∈aff(SB).(3)WecananalogouslydefinecrudelylinearfiniterepresentabilityofasetBinasetA.AfilterFisacollectionofsubsetsofasetΩsatisfyingi)∅∈F/;ii)A,B∈FimpliesA∩B∈F;iii)A∈FandA⊂B⊂ΩentailB∈F.AfilterFissaidtobefreeif∩{F∈F}=∅.AfilterUiscalledanultrafilterifforanyA⊂Ω,eitherA∈U,or,ΩA∈U.LetKbeatopologicalspace,andf:Ω→Kafunction.Wesayfisconvergenttosomek∈KwithrespecttoafilterFifforeveryneighborhoodUofk,wehavef−1(U)∈F;inthiscase,wedenotelimFf=k.Wewillrecallthedefinitionofanultraproduct(ultrapower)ofBanachspaces.ForanonemptysetΩ,let{Xω:ω∈Ω}beacollectionofBanachspaces.Thentheirultraproductisdefinedby∏⊕Xω=(Xω)ℓ1/{(xω):lim∥xω∥=0}.(4)UUω∈ΩlimU∥xω∥=0meansforallε>0,{ω∈Ω:∥xω∥<ε}∈U.Pleasenotethattheultraproductisaquotientoftheℓ∞-sumof{Xω},soitselementsareclassesoftherespectiveequivalencesrelation,notthegeneralizedsequencesitself.Wewilluseinthesequelthenotations[(xω)](or,simplyx=(x(ω)),ifnoconfusionarises)todenotetheequivalenceclassof(xω).Thus,foracollection{Aω⊂Xω:ω∈Ω}ofsubsets,itsultraproductis∏⊕Aω={[(xω)]:(xω)∈(Xω)ℓ1:xω∈Aωforω∈Ω}.(5)Uω∈Ω∏Inparticular,ifXω=X,andAω=A⊂Xforallω∈Ω,thenwedenotebyAU=UA,theU-ultrapowerofA.Fordistinction,wealsouse(x,y)Utodenotevectorsin(X×Y)Uand(xU,yU),vectorsinXU×YU.Proposition1.LetAbeanonemptysubsetofaBanachspaceX,andUafreeultrafilteronsomesetΩ.Theni)AUisfinitelyrepresentableinA;ii)AnylinearlyindependentsetB⊂AUislinearlyfinitelyrepresentableinA.Proof.Letx0=[(x0(ω))],···,xxn=[(xn(ω))]∈AUben+1vectors.Foreachjchooseanarbitraryrepresentant(xj(ω))withxj(ω)∈Aforallω∈Ω.Wefirstshowthatifx0,···,xxnarelinearlyindependentthenthesetU={ω∈Ω:x0(ω),···,xxn(ω)arelinearlyindependent}(6)-4- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn∑nbelongstoU.Otherwise,V≡ΩU∈U,andforeachω∈Vtheequationj=0αj(ω)xj(ω)=0∈Xhasanonzerosolution(α(ω),···,α(ω))∈Rn+1.Withoutlossofgeneralityweassume0n∑n∑n∑nj=0|αj(ω)|=1.Letαj=limUαj(ω)for0≤j≤n.Thenj=0|αj|=1andj=0αjxj=0∈XU.Thissaysthatx0,···,xxnarelinearlydependent,whichcontradictstothehypothesis.Analogously,wecanshowthatifthevectorsx0,···,xxnareaffinelyindependentthentheset′U={ω∈Ω:x0(ω),···,xxn(ω)areaffinelyindependent}(7)belongstoU.Indeed,bythefactwehavejustproven,itsufficestonote′U={ω∈Ω:(x1−x0)(ω),···,(xn−x0)(ω)arelinearlyindependent}.Wenextprovethatifx0,···,xxnarelinearlyindependentthentheset{x0,···,xxn}islinearlyfinitelyrepresentableinA.LetUbedefinedby(6).Thenforeachω∈U,Fω≡span{x0(ω),···,xxn(ω)}isann+1dimensionalsubspaceofX.ByapplyinganAuerbachbasis{e0(ω),···,en(ω)}ofFω(i.e.ej(ω)(j=0,···,n)areunitvectorsinFωsatisfyingthereexistn+1unitvectorse∗(ω)(j=0,···,n)inX∗suchthat⟨e∗(ω),e(ω)⟩=δ.Hereδ=0,ifjjiijiji̸=j;=1,otherwise),wecanshowthatthereisanorm|·|onRn+1withωn+1∥x∥∞≤|x|ω≤∥x∥1,∀x∈R(8)suchthatFislinearlyisometrictoRn+1≡(Rn+1,|·|)(see[16,Lemma3.3])Foreachω∈U,ωωωletL:F→Rn+1bealinearisometry.SinceU∈U,Arzelá-Ascolitheoremand(7)entailωωωthatthereexistsanorm|·|onRn+1suchthatlim|·|=|·|uniformlyoneachboundedUωsubsetofRn+1(seealso[16,Lemma3.4]).Consequently,F≡span{x,···,xx}∼=ΠE∼=ΠG∼=(Rn+1,|·|),(9)0nUωUωwhereE=F,ifω∈U;={0},otherwise;andG=Rn+1,ifω∈U;={0},otherwise.ωωωω∑n∑nForeachω∈U,letPω:F→FωbedefinedbyPω(j=0λjxj)=j=0λxxj(ω)forallλj∈R.ItisclearthatlimU∥Pω(x)∥=∥x∥forallx∈F,andPω(S)=Sω,whereS=co{x0,···,xxn}andSω=co{x0(ω),···,xxn(ω)}.Therefore,limU|Lω◦Pω(x)|ω=∥x∥forallx∈F,whichimpliesthatL◦PconvergestosomelinearisometryF→(Rn+1,|·|)alongωωU.Thus,PωconvergestosomelinearisometryP:F→XalongU.Consequently,foreveryε>0,thereexistsω∈Usuchthat(1−ε)∥x∥≤∥Pω(x)∥≤(1+ε)∥x∥,forallx∈F(10)withPω(S)=Sω.Therefore,wehaveshownthateverylinearlyindependentsubsetB⊂AUislinearlyfinitelyrepresentableinA,thatis,ii)holds.Toshowi),supposethatx0,···,xxnareaffinelyindependent.Wewriteyj=xj−x0.Thenthenvectorsy1,···,yynarelinearlyindependent.Let′F=span{y1,···,yyn}andF=aff{x0,···,xxn}.(11)-5- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnThenF′=x+F.Foreachω∈U′,let0′Fω=span{y1(ω),···,yyn(ω)},Fω=aff{x0(ω),···,xxn(ω)}(12)(whereyj(ω)=xj(ω)−x0(ω))anddefine′′′Pω:F→Fω,Pω:F→Fω(13)by∑n∑n∑nPω(λjyj)=λjyj(ω)forally=λjyj∈F(14)j=1j=1j=1and∑n∑n∑n′′Pω(λjxj)=λjxj(ω)forallx=λjxj∈F.(15)j=0j=0j=0ThenwehavelimU∥Pω(y)∥=∥y∥forally∈F.Note∑n∑n′Pω(λjxj)=x0(ω)+Pω(λjyj)(16)j=0j=1∑n′forallx=j=0λjxj∈F.Thisimplies′′′lim∥Pω(x)−Pω(y)∥=∥x−y∥forallxx,yy∈F.(17)UHence,wehaveshownthatforeveryε>0thereexistω∈U′,naffinelyindependentvectorsx(ω),···,x(ω)∈AandanaffinemappingP′:F′→F′suchthat0nωω′′′(1−ε)∥x−y∥≤∥Pω(x)−Pω(y)∥≤(1+ε)∥x−y∥forallxx,yy∈FandsuchthatP′(x)=x(ω)forj=0,···,n.ThissaysthatAisfinitelyrepresentableinωjjUA,i.e.i)holds.Proposition2.LetA⊂XandB⊂YbetwosubsetsofBanachspacesXandY,andAbounded.i)IfBisfinitelyrepresentableinA,thenforanyaffinelyindependentsubsetB0⊂BthereexistafreeultrafilterU,andanaffineisometryT:aff(B)→aff(A)UsuchthatT(B0)⊂AU.ii)IfBiscrudelyfinitelyrepresentableinA,thenforanyaffinelyindependentsubsetB0⊂BthereexistafreeultrafilterU,andanaffineembeddingT:aff(B)→aff(A)UsuchthatT(B0)⊂AU.Proof.i)SupposethatBisfinitelyrepresentableinA.ForeveryfamilyC⊂Bconsistingoffinitelymanyaffinelyindependentvectors,N≡co(C)isasimplexcontainedinthefinitedimensionalaffinespaceF≡aff(C).SinceBisfinitelyrepresentableinA,foreachm∈N,-6- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnthereexistasimplexMm≡co(Am)withvertexsetAm⊂A,andanaffinemappingT˜m,C:F→Em≡aff(Am)withT˜m,C(N)=Mmsuchthat(1−1/m)∥x−y∥≤∥T˜m,Cx−T˜m,Cy∥≤(1+1/m)∥x−y∥,∀x,y∈F.(18)ByZorn’slemma,foreachaffinelyindependentsubsetB0,thereexistsamaximalaffinelyindependentsubsetofBcontainingB0.Withoutlossofgenerality,wecanassumeB0itselfismaximal.Clearly,aff(B0)=aff(B).LetΩ=N×{C⊂B0:Cisfinite},andfixz0∈A.Further,letTm,C:aff(B)→aff(A)bedefinedasT˜m,Conaff(C)andasz0onaff(B)aff(C).NowfixanultrafilterUonΩwhichcontainsthesets′′′′{(m,C)∈Ω:m≥m,C⊃C}:(m,C)∈Ω,anddefineT:aff(B)→aff(A)UbyT(x)=[(Tm,C(x))(m,C)∈Ω].SinceAisbounded,thegeneralizedsequence{Tm,C(x)}(m,C)∈Ωisboundedforeachx.BythechoiceofU,itiseasytocheckthatTisanaffineisometryandthatT(B0)⊂AU.ii)SupposethatBiscrudelyfinitelyrepresentableinA.LetβbetheconstantfromDefinition1iii),letB0beasaboveandz0∈Abefixed.LetΩbethefamilyofallthefinitesubsetsofB0.ForC∈ΩletT˜C:aff(C)→aff(A)beprovidedbythedefinitionof“crudelyfinitelyrepresentable”.ExtendT˜CtoTCbyusingz0asabove.LetUbeanultrafilteronΩcontainingallthesetsoftheform{C′∈Ω:C′⊃C},whereC∈Ω.Finally,defineT:aff(B)→aff(A)UbyT(x)=[(TC(x))C∈Ω].ThenTisanaffineembeddingofaff(B)intoaff(A)UwithT(B0)⊂AU.2Somebasicpropertiesofnon-convexsuperweaklycompactsetsInthissection,weshallshowthatsuperweakcompactnessisinvariantunderoperationsofcontinuouslinearmapping,productandunionoffinitelymanysetsandn-convexcombinationsofaset.Definition2.A(weaklyclosed,resp.)subsetAofaBanachspaceXissaidtoberelativelysuperweaklycompact(superweaklycompact,resp.)providedeverysubsetBofaBanachspaceYwhichisfinitelyrepresentableinAisrelativelyweaklycompact.Thisconceptincorporatingof[13,Corollary2.15]entailsthatinaBanachspaceXeverycompactsetissuperweaklycompact;Xissuper-reflexiveifandonlyifitsclosedunitballissuperweaklycompact;andifXissuperreflexivetheneveryboundedsetisrelativelysuperweaklycompact.-7- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnAswehavewellknown,theweakclosureofasubsetinaninfinitedimensionalBanachspacecouldbemuchlargerthanthesubset.Toavoiddifficultyof“weakclosure”,weshalluse“relativesuperweakcompactness”insteadof“superweakcompactness”inthefollowingdiscussion.Weshallseethatasubsetisrelativelysuperweaklycompactisequivalenttotheweakclosureofthesubsetissuperweaklycompact(Proposition5).Proposition3.LetΩbeaset,{X,Xω:ω∈Ω}beacollectionofBanachspaces,A⊂X,Aω⊂Xωforallω∈Ω,andletUbeafreeultrafilteronΩsatisfyingthatthereisasequence{Un}⊂Usothat∩Un=∅;inparticular,afreeultrafilteronN.Then∏∏i)UAωisnormclosedinUXω;ii)ThereisT∈B(X,X∗∗)with∥T∥=1sothatUa)T◦D=Id,theidentity;wwb)s-A⊂T(AU);Inparticular,ifAisrelativelyweaklycompact,thenA⊂T(AU),wherews-AistheweaklysequentialclosureofA,andDisthenaturaldiagonalidentityembeddingfromXtoXU.∏∏Proof.i).Givenx∈A,let{x}⊂Abeasequencewith∥x−x∥<1.FixsomeUωnUωnnrepresentants(x(ω))and(x(ω))ofxand{x}.ThenV≡{ω∈Ω:∥x(ω)−x(ω)∥<1}∈U.nnnnnSetWn=Un∩V1∩···∩Vn.ThenWn∈U.Fixanyx0∈Aandlety=[(y(ω))ω∈Ω]∈AUbe∏definedbyy(ω)=xm(ω),ω∈WmWm+1,m∈N;=x0,otherwise.Clearly,x=y∈UAω.ii)NotethatifUisanultrafilterandifKisacompactHusdorfftopologicalspacethenforanyf:Ω→Kthereisauniquek∈KsothatlimUf=k.LetXbeendowedwiththew∗-topologyofX∗∗anddefineT(x)=w∗-limx(ω)forallx∈X.ThenweseethatUUT:X→X∗∗isalinearoperatorofnormone,andsatisfiesT◦D=Id,thatis,a)holds.UToshowb),let{xn}⊂Abeasequencesuchthatxn→x∈XintheweaktopologyofX,andlet{Un}⊂Ubeamonotonenon-increasingsequencesothat∩Un=∅.Fixanyx0∈Aanddefinex=[(x(ω))ω∈Ω]∈AUbyx(ω)=xm,ω∈UmUm+1,m∈N;=x0,otherwise.Thenwwwwehavex=T(x).Thus,s-A⊂T(AU).Finally,itsufficestonotes-A=AwheneverAisrelativelyweaklycompact.Proposition4.LetAbeanonemptysubsetofaBanachspaceX.Thenthefollowingstatementsareequivalent.i)Aisrelativelysuperweaklycompact;ii)AUisrelativelysuperweaklycompactforsomefreeultrafilterU;iii)AUisrelativelysuperweaklycompactforeveryfreeultrafilterU;iv)AUisrelativelyweaklycompactforeveryfreeultrafilterU.Proof.i)=⇒iii).SinceAisrelativelysuperweaklycompact,itisbounded.ByProposition1,AUisfinitelyrepresentableinA.Thus,transitivityoffiniterepresentabilityimpliesthatifasubsetBofaBanachspaceisfinitelyrepresentableinAU,thenitisfinitelyrepresentablein-8- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnA.RelativesuperweakcompactnessofAentailsthatBisrelativelyweaklycompact.Hence,AUisrelativelysuperweaklycompactforeveryfreeultrafilterU.iii)=⇒ii)=⇒i)andiii)=⇒iv)aretrivial.Itremainstoshowiv)=⇒i).Clearly,Aisbounded.Suppose,tothecontrary,thatAisnotrelativelysuperweaklycompact.Then,thereisaboundednon-relativelyweaklycompactsetB,whichisfinitelyrepresentableinA.Consequently,thereisasequence{xn}⊂Bwhichhasnoweakclusterpoint.Wecanassumethat{xn}isalinearlyindependentset.ByProposition2,thereexistafreeultrafilterUandanaffineisometryT:aff(B)→aff(A)Usuchthat{T(xn)}⊂AU.Clearly,{T(xn)}isnotrelativelyweaklycompact.ThisisacontradictiontorelativeweakcompactnessofAU.Corollary1.TheunionoftworelativelysuperweaklycompactsetsinaBanachspaceisagainrelativelysuperweaklycompact.Proof.LetA,BbetworelativelysuperweaklycompactsetsofaBanachspaceX.Suppose,tothecontrary,thatC=A∪Bisnotrelativelysuperweaklycompact.ByProposition4,thereisafreeultrafilterUsuchthatCUisnotrelativelyweaklycompact.AgainbyProposition3.4,bothAUandBUarerelatively(super)weaklycompact.Consequently,CUisrelativelyweaklycompactsinceCU=AU∪BU.Thisisacontradiction.Theorem1.LetA⊂XandB⊂YbetwoboundedsubsetsofBanachspacesXandY.SupposethatT:X→YisaboundedlinearoperatorwithT(A)=B.IfAisrelativelysuperweaklycompact,thenBisalsorelativelysuperweaklycompact.Proof.SupposethatBisnotrelativelysuperweaklycompact.ThenbyProposition4,thereisafreeultrafilterUsuchthatBUisnotrelativelyweaklycompactinYU.SinceAisrelativelysuperweaklycompact,againbyProposition4,AUisrelativelysuperweaklycompact.WedefineS:XU→YUbyS(x)=[(Txx(ω))ω∈Ω].ThenSisaboundedlinearoperatorwithS(AU)=BU.ThiscontradictstothatS(AU)isnotrelativelyweaklycompact.Corollary2.i)Relativesuperweakcompactnessisinvariantunderlinearisomorphisms;ii)Aboundedsetcrudelyfinitelyrepresentableinarelativelysuperweaklycompactsetisagainrelativelysuperweaklycompact.Proof.i)isjustadirectconsequenceofTheorem1.Toshowii),letX,YbeBanachspaces,A⊂Xbearelativelysuperweaklycompactset,andB⊂YbecrudelyfinitelyrepresentableinA.IfBisnotrelativelysuperweaklycompact,thenbyProposition3thereisanon-relativelysuperweaklycompactsequenceB0≡{xn}⊂Boflinearlyindependentvectors.By-9- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnProposition2,thereisanaffineembeddingT:aff(B)→aff(A)UsuchthatT(B0)⊂AUforsomefreeultrafilterU,andAUisrelativelysuperweaklycompact.Fixb0∈BandletS(x)=T(b)−T(b0)foreachx=b−b0∈aff(B)−b0≡F.ThenSisalinearembeddingfromthelinearspaceFtothelinearspaceE=aff(A)U−T(b0)satisfyingS(B−b)⊂A−T(b).SinceB−b=S−1[S(B−b)]isnotrelativelysuperweakly00U00000compact,byTheorem1,S(B0−b0)isnotrelativelysuperweaklycompact.ThiscontradictstothatS(B0−b0)isasubsetoftherelativelysuperweaklycompactsetAU−T(b0).Theorem2.Theproductoftworelativelysuperweaklycompactsetsisagainrelativelysuperweaklycompact.Proof.LetA⊂XandB⊂YbetworelativelysuperweaklycompactsetsofBanachspacesXandY.Weputℓ1normontheproductZ≡X×Y,i.e.∥(x,y∥=∥x∥+∥y∥forall(x,y)∈Z.Suppose,tothecontrary,thatC≡A×Bisnotrelativelysuperweaklycompact,thenbyProposition4again,thereisafreeultrafilterUsuchthatCUisnotrelativelyweaklycompact.NotethatT:ZU→XU×ℓ1YUdefinedbyT((x,y)U)=(xU,yU)isalinearisometry,andnotethatbothAUandBUarerelativelyweaklycompact.WeseeT(CU)=AU×BUisrelativelyweaklycompact.Thus,CUisrelativelyweaklycompact,andthisisacontradiction.Corollary3.LetAandBbetworelativelysuperweaklycompactsubsetsofaBanachspaceX.ThenA+Bisagainrelativelysuperweaklycompact.Proof.ByTheorem2,A×BisrelativelysuperweaklycompactinX2.SinceT:X2→XdefinedbyT(x,y)=x+yisaboundedlinearoperator,duetoTheorem1,T(A×B)=A+Bisrelativelysuperweaklycompact.Proposition5.AsubsetAofaBanachspaceisrelativelysuperweaklycompactifandonlywifAissuperweaklycompact.Proof.Itsufficestoshownecessity.SinceAisrelativelysuperweaklycompact,foranyfreeultrafilterUonN,byProposition4,AUisagainrelativelysuperweaklycompact.MakingusewofProposition3ii)b),thereisaboundedlinearoperatorTsuchthatA⊂T(AU).Theoremw1entailsAissuperweaklycompact.3CharacterizationsofsuperweaklycompactsetsInthissection,withtheaimofJames’characterizationofnon-weaklycompactsets[17],andpropertiesestablishedintheprevioussections,weshallshowtwocharacterizationsofarelativelysuperweaklycompactset.OneisthefollowingtheoremofGrothendiek’stype;andtheotherisJames’typecharacterizationfornon-superweaklycompactsets(Theorem4).-10- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnTheorem3.AsubsetAofaBanachspaceXisrelativelysuperweaklycompactif(andonlyif)foreveryε>0thereisarelativelysuperweaklycompactsetBsuchthatA⊂B+εBX.Proof.Suppose,tothecontrary,thatAisnotrelativelysuperweaklycompact.ThenbyProposition4,thereisafreeultrafilterUsuchthatAUisnotrelativelyweaklycompact,whileBUisrelativelyweaklycompact.NoteAU⊂(B+εBX)U=BU+εBXU.ByGrothendieck’slemma,AUisrelativelyweaklycompact.Thiscontradictioncompletesourproof.Problem.ForarelativelysuperweaklycompactsetA,wedonotknowwhetherco(A)isagainrelativelysuperweaklycompact.Theorem4.LetAbeanonemptyboundedweaklyclosedsubsetofaBanachspaceX.Thenitisnotsuperweaklycompactifandonlyifithasuniformlyfinitebi-orthogonalproperty,i.e.∗∗thereisθ>0sothatforalln∈Nthereexistnvectorsx1,···,xn∈Aandx1,···,xn∈BXsatisfying{θ,1≤i≤j≤n;∗⟨xi,xj⟩=(19)0,1≤j