the unconditional basic sequence problem

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1、THEUNCONDITIONALBASICSEQUENCEPROBLEMW.T.GowersandB.Maurey-preliminaryversionAbstract.WeconstructaBanachspacethatdoesnotcontainanyinfiniteunconditionalbasicsequence.§0.Introduction.AfundamentalroleinthetheoryofBanachspacesisplayedbythenotionofaSchauderbasis.IfXisaBanachspace,thenase

2、quence(x)∞isaSchauderbasisnn=1∞P∞(orsimplyabasis)ofXiftheclosedlinearspanof{xn}n=1isXandn=1anxniszeroonlyifeachaniszero.Thesecondcondition,assertingakindofindependence,clearlydependsverymuchontheorderofthexn,anditiscertainlypossibleforapermutationofabasistofailtobeabasis.Ontheothe

3、rhand,manybasesthatoccurnaturally,suchasthestandardbasisofℓpwhen1≤p<∞,arebasesunderanypermutation.Itisthereforenaturaltogiveanametothisspecialkindofbasis.Asithappensthereareseveralequivalentdefinitions.Theorem0·1.LetXbea(realorcomplex)Banachspaceandlet(x)∞beabasisnn=1ofX.Thenthefol

4、lowingareequivalent.(i)(x)∞isabasisofXforeverypermutationπ:N→N.π(n)n=1P∞(ii)Sumsoftheformn=1anxnconvergeunconditionallywhenevertheyconverge.(iii)ThereexistsaconstantCsuchthat,foreverysequenceofscalars(a)∞andeverynn=1sequenceofscalars(ǫ)∞ofmodulusatmost1,wehavetheinequalitynn=1X∞X∞

5、arXiv:math/9205204v1[math.FA]6May1992ǫnanxn≤Canxn.n=1n=1Abasissatisfyingtheseconditionsiscalledanunconditionalbasis,andabasissatis-fyingthethirdconditionforsomegivenconstantCiscalledC-unconditional.Aninfinitesequencethatisabasisofitsclosedlinearspaniscalledabasicsequence:ifitisanun

6、conditionalbasisofitsclosedlinearspanthenitisanunconditionalbasicsequence.ForalongtimeamajorunsolvedproblemwaswhethereveryseparableBanachspacehadabasis.ThisquestionwasanswerednegativelybyP.Enfloin1973[E].Ontheother1hand,itisnothardtoshowthateveryspacecontainsabasicsequence.Spaceswi

7、thunconditionalbaseshavemuchmorestructurethangeneralspaces,soexamplesofspaceswithoutthemareeasiertofind.Indeed,thespacesC([0,1])andL1donothaveanuncon-ditionalbasis.Thisleavesthequestionofwhethereveryspacecontainsanunconditionalbasicsequence,orequivalentlyhasaninfinite-dimensionalsub

8、spacewithanunconditionalbasis.The