steinhaus' lattice-point problem for banach spaces 2017 tomasz kania参考.pdf

该【steinhaus' lattice-point problem for banach spaces 2017 tomasz kania参考 】是由【小舍儿】上传分享，文档一共【11】页，该文档可以免费在线阅读，需要了解更多关于【steinhaus' lattice-point problem for banach spaces 2017 tomasz kania参考 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。:..JID:YJMAAAID:20735/FLADoctopic:FunctionalAnalysis[m3L;;Prn:22/09/2016;16:36](1-11).???(????)???–???ContentslistsavailableatScienceDirectJournalofMathematicalAnalysisandApplicationsate/jmaaSteinhaus’lattice-pointproblemforBanachspaces?TomaszKaniaa,b,TomaszKochanekc,d,?aSchoolofMathematicalSciences,WesternGatewayBuilding,UniversityCollegeCork,Cork,IrelandbMathematicsInstitute,UniversityofWarwick,GibbetHillRd,Coventry,CV47AL,UnitedKingdomcInstituteofMathematics,PolishAcademyofSciences,?niadeckich8,00-656Warsaw,PolanddInstituteofMathematics,UniversityofWarsaw,Banacha2,02-097Warsaw,PolandarticleinfoabstractArticlehistory:Steinhausprovedthatgivenapositiveintegern,onemay?,whoreplacedtheintegerlatticebyanyin??:spacessatisfyingthisproperty,whichwecall(S),andcharacterisethembymeansSteinhaus’problemofanewgeometricpropertyoftheunitspherewhichallowsustoshow,.,thatallLatticepointsstrictlyconvexnormshave(S),heless,thereareplentyofnon-strictlyconvexStrictlyconvexspacenormssatisfying(S).??rstobservedbySteinhaus[10,]:foranynaturalnumbernonemay?[11]-?niteset,.,anin?nitesubsetAofametricspaceXsuchthateachballinXcontainsonly?([11]).LetAbeaquasi-??Xsuchthatforeveryy∈Yandn∈NthereexistsaballBcentredatywith|A∩B|=’property(S).AmetricspaceXhasthispropertyif,byde?nition,?The?rst-namedauthoracknowledgeswiththanksfundingreceivedfromtheEuropeanResearchCouncil/-namedauthorwassupportedbythePolishMinistryofScienceandHigherEducationintheyears2013–14,.*-mailaddresses:@,t.******@(),******@().http://dx./.-247X/CrownCopyright?.:..JID:YJMAAAID:20735/FLADoctopic:FunctionalAnalysis[m3L;;Prn:22/09/2016;16:36](1-11),..???(????)???–???(S)foranyquasi-?nitesetA?XthereexistsadensesetY?Xsuchthatforally∈Yandn∈NthereexistsaballBcentredatywith|A∩B|=(S),formulatedabove,,theyrequirethat,locally,?cial,asitwillallowustoidentifyspacesthatsharethatpropertywithHilbertspaces,yetofaverydi??:(S)XhasSteinhaus’property;(S1)foranyquasi-?nitesetA?XthereexistsadensesetY?Xsuchthatforeveryy∈YthereexistsaballBcentredatywith|A∩B|=1;(S’)forallx,y∈Xwithx=y,x=y=1andeachδ>0thereexistsaz∈Xwithz<δsuchthatoneofthevectorsx+zandy+zhasnormgreaterthan1,whereastheotherhasnormsmallerthan1;(S”)forallx,y∈Xwithx=y,x=y=1andeachδ>0thereexistsaz∈Xwithz<δsuchthatx+z=y+z.Inotherwords,condition(S”)meansexactlythatonecannot?nda‘neighbourhood’,aswewillsee,,incontrasttomanyotherclassicalproperties,property(S)isnotinheritedbysubspacesand,inasense,(S’)and(S”)arerelatedtoanother(weaker)propertyof‘non-?atness’oftheunitsphere:(F)theunitsphereSXofXdoesnotcontainany?atfaces,thatistosay,thereisnonon-emptysubsetofSX,openintherelativenormtopology,,.,asetoftheformx+ker(x?)forsomex∈Xandx?∈X?.Note,however,that(F)doesnotimply(S”)thatis3witnessedbythenorm(x,y,z)=max{x2+y2,|z|}for(x,y,z)∈R(considerthepoints(1,0,0)and(1,0,1)).However,whethereveryBanachspaceadmitsarenormingsatisfying(F)’sresulttostrictlyconvexBanachspaces(Corollary2).Itiswell-knownthatnoteveryBanachspaceadmitsastrictlyconvexrenorming,justtomentiontheexamplesof∞(Γ)foranyuncountablesetΓ(see[3]and[4,§])orthequotientspace∞/c0([2]).Thismotivatesthequestionofwhetherstrictconvexityandproperty(S)areequivalentatthelevelofrenormings,-valuedmeasurablecardinal,thereexistsanon-strictlyconvexifableBanachspacewhosenormsatis?es(S).Moreover,foranyBanachspaceXwehave:(i)ifdimX2,thenXhasproperty(S)ifandonlyifXisstrictlyconvex;(ii)ifdimX>2andXadmitsarenormingwithproperty(S),thenitalsoadmitsanon-strictlyconvexrenormingwithproperty(S).:..JID:YJMAAAID:20735/FLADoctopic:FunctionalAnalysis[m3L;;Prn:22/09/2016;16:36](1-11),..???(????)???–???3Solovay([9])provedthattheassertionthatthecontinuumisareal-valuedcardinalisequiconsistentwiththeexistenceofatwo-valuedmeasurablecardinalnumber,thereforeitsconsistencycannotbeprovedinZFCalone(assumingofcoursethatZFCitselfisconsistent).Interestingly,ourconstructioninthisuniverseispossiblebecausethereal-valuedmeasurabilityofthecontinuumimpliesthefailureoftheContinuumHypothesis([1,])andwetakeadvantagenotonlyofpleasantmeasure--measurabilityofthecontinuumisreallynecessarytoshowthatthereexistBanachspaceswith(S)(S)?(S1)and(S’)?(S”)holdtruetrivially,itisenoughtoprovethat(S1)?(S’),(S’)?(S)and(S”)?(S’).(S1)?(S’):Supposethat(S1)>0andx,y∈Xwithx=y,x=y=-?nitesetA?XsuchthatA∩(1+δ)BX={x,y},(S1),thereisau∈X,u<δ/2,suchthatforsomer>0theopenballB(u,r)∈A\{x,y}belongingtoB(u,r).Thenδδr>a?ua?u>(1+δ)?=1+,22hencex?u<r,thatisx∈B(u,r);,B(u,r)containsexactlyoneofthepointsxandy,sayx∈B(u,r)andy/∈B(u,r).Thenδδ1?<x?u<ry?u<1+.22Supposethatr1,r=1?εwithsomeε∈[0,δ/2)andtakeanynumberρsatisfyingδ0<ρ<minr?x?u,?,wemay?ndv∈Xwithvε+ρsuchthaty?(u+v)r+ε+ρ>x?(u+v)x?u+v<r?ρ+v,settingz=?(u+pletestheproofofourclaim,sincewehavetheestimateu+v<ε+ρ+δ/2<>1sotheproofof(S1)?(S’)plete.(S’)?(S):LetXbeaBanachspaceXthatsatis?es(S’)andletA?Xbeaquasi-?∈NsetGn=x∈X:|A∩B(x,r)|=nforsomer>,inviewofthede?nitionofaquasi-?niteset,,insearchofacontradiction,thatthereisanopenballU=B(x0,r0),wemaysupposethatA∩U=?.Withanypointx∈Uweassociatetwointegersm(x)<nandk(x)2de?nedasfollows:Sincex∈/Gn,thereisthelargestnon-negativeinteger:..JID:YJMAAAID:20735/FLADoctopic:FunctionalAnalysis[m3L;;Prn:22/09/2016;16:36](1-11),..???(????)???–???m(x)<nforwhichthereexistsq>0with|A∩B(x,q)|=m(x).Then,foreverys>qwehaveeither|A∩B(x,s)|=m(x)or|A∩B(x,s)|>?nes=inft>0:|A∩B(x,t)|>(x)pointsa1,...,am(x)∈AlieintheballB(x,s),whereasatleasttwosuchpointslieontheboundaryofB(x,s);letuscallthemb1,...,bk,wherek?nek(x)=,weshalluseanin?,...,am,b1,...,bkbeasaboveforx=x0,wherem=m(x0)andk=k(x0).Pickanyδ>0suchthat{ai:1im}?B(x0+u,s)∩A?{ai,bj:1im,1jk}foreveryu∈Xwithu<?neρ=max{ai?x0:1im}<sandsetγ=s?(bj?x0)/s(j=1,...,k)(S’)toanytwoofthem(.,toj=1,2),weobtainapointz∈Xwithsz<min{δ,γ/2}suchthatoneofthevectors:bj?x0?sz(j=1,2)hasnormgreaterthans,,ifnecessary,wemayalsoassumethatthepointx:=x0+(x,s)withthecentreinUcontainsallai’s(1im)andatleastonebutnotallamongbj’s(1jk).Observealsothatbyourchoiceofz,wehaveai?xai?x0+sz<ρ+γ/2=s?γ/2foreach1imandbj?xbj?x0?sz>s?γ/2foreach1j,bysuitablyrescalingtheballB(x,s),weobtainanewballcentredatxwhichcontainsallofai’sandwhoseboundarycontainssomebutnotallofbj’(x)>m(x0)ork(x)<k(x0).Thisconstruction(withx0replacedbyx)willultimatelyleadtoacontradiction,aswe?nallyarriveatapointu∈Uwithm(u)nork(u)<,allthesetsGn(n∈N)areopenanddense.BytheBaireCategoryTheorem,thesetY=∞GisdenseinXand,obviously,foreachy∈Yandn=1nn∈NthereisaballBcentredatywith|A∩B|=(S).(S”)?(S’):Assumethenegationof(S’)andchoosedistinctunitvectorsx,y∈Xandδ>0sothatthereisnovectorz∈Xwithz<δforwhichexactlyoneofthevectorsx+zandy+∈SXde?neVu=z∈SX:u+αz<1forsomeα>0andλu(z)=minδ,inf{α>0:u+αz1}(z∈Vu).Bytheassumption,wehaveVx=Vyandλx(z)=λy(z)foreveryz∈Vx,whichmeansthattheunitspherelookslocallythesameatxandy(viathetranslationbyy?x),namely,:..JID:YJMAAAID:20735/FLADoctopic:FunctionalAnalysis[m3L;;Prn:22/09/2016;16:36](1-11),..???(????)???–???5y?x+(B(x,δ)∩SX)=B(y,δ)∩SX.(1)Pickη>0sosmallthatx+zy+zx?<δandy?<δifz<η.(2)x+zy+zNow,using(S”),chooseavectorz∈Xwithz<ηsothatx+z=y+z.Wehavethentwopossibilities:eitherx+z1andy+z1,orx+z1andy+z;x+z>y+z.Considerthefunctiong:[0,∞)→[0,∞)givenbyg(α)=x+z+α(y?x)whichisconvex,ascanbeeasilyveri?(1)and(2),wehavex+zy?x+=1,x+zthatis,g(x+z)=x+z.Wehavealsog(0)=x+zandg(1)=y+z<x+z.Thisisacontradictionwiththeconvexityofg,asthearguments:0,x+zand1lieinthisorderontherealline.’,yinarealvectorspaceX,wedenotebyxythelinesegmentbetweenxandy,.,xy={λx+(1?λ)y:λ∈[0,1]}.,y∈SX,thenforeachδ>0thereisz∈Xwithz<δsuchthatoneofthevectorx+z,y+>0andx,y∈Xwithx=y,x=y=,joiningxandy,hasnormsmallerthan1,whereaseachpointlyingonthestraightlinepassingthroughxandy,butoutsidexy,,anypointz∈Xsatisfying0<z<δandx+z∈xydoesthejob.?es(S).Now,wewillseethatstrictlyconvexspacesdonotexhaustthewholeclassofBanachspacessatisfyingSteinhaus’,thesetwoclassesdi?|||·|||inR3suchthat(R3,|||·|||)contains2isometrically∞(andhenceisnotstrictlyconvex),helessitsatis?escondition(S).Weareindebtedtotherefereeforsuggestingthefollowingexamplewhichsigni?cantlysimpli?edouroriginalconstruction.:..JID:YJMAAAID:20735/FLADoctopic:FunctionalAnalysis[m3L;;Prn:22/09/2016;16:36](1-