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1、ANALYTICSHEAVESINBANACHSPACESLaszl´oLempert´Abstract.WeintroduceaclassofanalyticsheavesinaBanachspaceX,thatweshallcallcohesivesheaves.Cohesionismeanttogeneralizethenotionofcoherencefromfinitedimensionalanalysis.Accordingly,weprovetheanalogofCartan’sThe-oremsAandBforc
2、ohesivesheavesonpseudoconvexopensubsetsΩ⊂X,providedXhasanunconditionalbasis.IntroductionInfinitedimensionalcomplexanalysisandgeometrycoherentanalyticsheavesplayacentralrole,forthefollowingfourreasons:(i)Mostsheavesthatoccurinthesubjectarecoherent.(ii)Overpseudoconvex
3、subsetsofCntheirhighercohomologygroupsvanish.(iii)Theclassofcoherentsheavesisclosedundernaturaloperations.(iv)Whetherasheafiscoherentcanbedecidedlocally.Thepurposeofthispaperistointroduceacomparableclassofsheaves,thatweshallcallcohesive,inBanachspaces.Thisnotionisdi
4、fferentfromcoherence,whichformallymakessenseininfinitedimensionsaswell.However,coherenceisnotrelevantforinfinitedimensionalgeometry,sinceithastodowithfinitegeneration,whileininfinitedimensionalspacesonefrequentlyencounterssheaves,suchastangentsheavesandidealsheavesofpoin
5、ts,thatarenotfinitelygeneratedoverthestructuresheaf.ThestructuresheafitselfisnotknowntobecoherentinanyinfinitedimensionalBanachspace,either.WeshalldefinecohesivesheavesingeneralBanachspaces(alwaysoverC).How-arXiv:math/0507549v1[math.CV]26Jul2005ever,weareabletoprovemea
6、ningfulresultsaboutcohesivesheavesonlyinsomeBanachspaces,e.g.inthosethathaveanunconditionalbasis.(Ourmainresultsdonotgeneralizetocertainnonseparablespaces,andwedonotknowwhethertheyholdinallseparablespaces,oratleastinthosethathaveaSchauderbasis.—ForthenotionofSchaude
7、randunconditionalbases,seeSection1.)Cohesivesheavesaresheavesofmoduleswithanextrastructureandaspecialproperty.Weshallarriveattheirdefinitioninfoursteps.GivenaBanachspaceResearchpartiallysupportedbyNSFgrantDMS02030721991MathematicsSubjectClassification.32C35,46G20,32T.
8、TypesetbyAMS-TEX12LASZL´OLEMPERT´X,anopenΩ⊂X,andanotherBanachspaceE,thesheafOE=OEofgermsofΩholomorphicfunctions(Ω,x)→E,x∈Ω,willbecalledapl
