Chapter The Uniform Boundedness Principle and the Clo - PDF document

Chapter2TheUniformBoundednessPrincipleandtheClosedGraphTheorem2.1TheBaireCategoryTheoremThefollowingclassicalresultplaysanessentialroleintheproofsofChapter2.Theorem2.1(Baire).beacompletemetricspaceandletbeasequenceofclosedsubsetsinAssumethatforeveryTheBairecategorytheoremisoftenusedinthefollowingform.Letbeanonemptycompletemetricspace.LetbeasequenceofclosedsubsetssuchthatThenthereexistssomesuchthatIntProof.,sothatisopenanddenseinforevery1.Ouraimistoprovethatisdensein.Letbeanonemptyopensetin;weshallprovethatAsusual,setB(x,r)d(y,x)rPickany0suchthat B(xThen,chooseB(x0suchthatFunctional Analysis, Sobolev Spaces and Partial Differential Equations, 2TheUniformBoundednessPrincipleandtheClosedGraphTheorem B(xB(x whichisalwayspossiblesinceisopenanddense.Byinductiononeconstructstwosequencessuchthat B(xB(x ItfollowsthatisaCauchysequence;letB(xforevery0andforevery0,weobtainatthelimit(as B(xInparticular,2.2TheUniformBoundednessPrinciplebetwon.v.s.Wedenoteby(E,F)thespaceofoperatorsfromequippedwiththenorm(E,F)Asusual,onewrites(E)insteadof(E,E)Theorem2.2(Banach…Steinhaus,uniformboundednessprinciple).betwoBanachspacesandletbeafamilynotnecessarilycountablecontinuouslinearoperatorsfrom.Assumethat(E,F)Inotherwords,thereexistsaconstantsuchthatTheconclusionofTheorem2.2isquiteremarkableandsurprising.FrompointwiseestimatesonederivesaProof.Forevery1,let 2.2TheUniformBoundednessPrinciplesothatisclosed,andby(1)wehaveItfollowsfromtheBairecategorytheoremthatIntforsome1.Pick0suchthatB(x,r).Wehaverz)Thisleadsto(E,F)whichimplies(2).Recallthatingeneral,apointwiselimitofcontinuousmapsneedcontinuous.ThelinearityassumptionplaysanessentialroleinTheorem2.2.Note,however,thatinthesettingofTheorem2.2itdoesfollowthat(E,F)Hereareafewdirectconsequencesoftheuniformboundednessprinciple.Corollary2.3.betwoBanachspaces.Letbeasequenceofcon-tinuouslinearoperatorsfromsuchthatforeveryconverges)toalimitdenotedby.Thenwehave(a)sup(E,F)(E,F),(E,F)liminf(E,F)Proof.(a)followsdirectlyfromTheorem2.2,andthusthereexistsaconstantsuchthatAtthelimitwe“ndisclearlylinear,weobtain(b).Finally,wehave(E,F)and(c)followsdirectly.Corollary2.4.beaBanachspaceandletbeasubsetof.Assumethatforeverythesetf(B)={f,xisboundedisbounded 2TheUniformBoundednessPrincipleandtheClosedGraphTheoremProof.WeshalluseTheorem2.2with,and.Forevery,set(f)f,bsothatby(3),(f)ItfollowsfromTheorem2.2thatthereexistsaconstantsuchthatf,b\f|Thereforewe“nd(usingCorollary1.4)thatCorollary2.4saysthatinordertoprovethatasetisboundeditsuf“cestolookŽatthroughtheboundedlinearfunctionals.Thisisafamiliarprocedurein“nite-dimensionalspaces,wherethelinearfunctionalsarethecomponentswithrespecttosomebasis.Insomesense,Corollary2.4replaces,inin“nite-dimensionalspaces,theuseofcomponents.Sometimes,oneexpressestheconclusionofCorollary2.4bysayingthatweaklyboundedŽstronglyboundedŽ(seeChapter3).NextwehaveastatementdualtoCorollary2.4:Corollary2.5.beaBanachspaceandletbeasubsetof.Assumethatforeverytheset\f={f,xisboundedisboundedProof.UseTheorem2.2with,and.Forevery(x)b,xE).We“ndthatthereexistsaconstantsuchthatb,x\f|Weconclude(fromthede“nitionofadualnorm)that2.3TheOpenMappingTheoremandtheClosedGraphTheoremHerearetwobasicresultsduetoBanach. 2.3TheOpenMappingTheoremandtheClosedGraphTheoremTheorem2.6(openmappingtheorem).betwoBanachspacesandbeacontinuouslinearoperatorfromthatissurjective.ThenthereexistsaconstantsuchthatT(B,c).Property(7)impliesthattheimageunderofanyopensetinisanopensetin(whichjusti“esthenamegiventothistheorem!).Indeed,letussupposeisopeninandletusprovethatT(U)isopen.FixanypointT(U),soforsome.Let0besuchthatB(x,r),i.e.,,r).ItfollowsthatT(B(,r))T(U).Using(7)weobtainT(B(,r)),rc)andthereforeB(y,rc)T(U).SomeimportantconsequencesofTheorem2.6arethefollowing.Corollary2.7.betwoBanachspacesandletbeacontinuouslinearoperatorfromthatisbijective,i.e.,injectiveandsurjective.isalsocontinuousfromE).ProofofCorollaryProperty(7)andtheassumptionthatisinjectiveimplythatischosensothat,then1.Byhomogeneity,we“ndthat andthereforeiscontinuous.Corollary2.8.beavectorspaceprovidedwithtwonorms,AssumethatisaBanachspacefornormsandthatthereexistsaconstantsuchthatThenthetwonormsare,i.e.,thereisaconstantsuchthatProofofCorollaryApplyCorollary2.7with(E,),F(E,ProofofTheoremWesplittheargumentintotwosteps:Step1.Assumethatisalinearsurjectiveoperatorfrom.Thenthereexistsaconstant0suchthat 2TheUniformBoundednessPrincipleandtheClosedGraphTheorem T(B(c).Proof. T(B(.Sinceissurjective,wehave,andbytheBairecategorytheoremthereexistssomesuchthatInt.Itfollows T(B(]\t0andsuchthatB(y T(B(Inparticular, T(B(,andbysymmetry,(10) T(B(Adding(9)and(10)leadsto T(B( T(B(Ontheotherhand,since T(B(isconvex,wehave T(B( T(B( T(B(and(8)follows.Step2.isacontinuouslinearoperatorfromthatsatis“es(8).ThenwehaveT(B(,c).Proof.Chooseany.Theaimisto“ndsomesuchthat1andBy(8)weknowthat .2,we“ndsomesuchthat 2and\nyŠTz1\nc Bythesameconstructionappliedto(insteadof)with4we“ndsuchthat 4and\n(yŠTz1)ŠTz2\nc Proceedingsimilarly,byinductionweobtainasequencesuchthat 2.4ComplementarySubspaces.RightandLeftInvertibilityofLinearOperators T(z+···+ Itfollowsthatthesequence+···+isaCauchysequence.Letwith,clearly,1andiscontinuous).Theorem2.9(closedgraphtheorem).betwoBanachspaces.Letbealinearoperatorfrom.AssumethatthegraphofG(T),isclosedin.Theniscontinuous.Theconverseisobviouslytrue,sincethegraphofanycontinuousmap(linearornot)isclosed.ProofofTheoremConsider,on,thetwonorms(thenormiscalledthegraphnormItiseasytocheck,usingtheassumptionthatG(T)isclosed,thatisaBanachspaceforthenorm.Ontheotherhand,isalsoaBanachspaceforthenorm\n\n.ItfollowsfromCorollary2.8thatthetwonormsareequivalentandthusthereexistsaconstant0suchthat.Weconcludethat2.4ComplementarySubspaces.RightandLeftInvertibilityofLinearOperatorsWestartwithsomegeometricpropertiesofclosedsubspacesinaBanachspacethatfollowfromtheopenmappingtheorem.Theorem2.10.beaBanachspace.Assumethataretwoclosedlinearsubspacessuchthatisclosed.ThenthereexistsaconstantsucheveryadmitsadecompositionoftheformG,yProof.Considertheproductspacewithitsnormnormx,y]\n=\nandthespaceprovidedwiththenormofThemappingde“nedbybyx,yiscontinuous,linear,andsurjective.Bytheopenmappingtheoremthereexistsaconstantsuchthateverycanbewrittenas,and1.Byeverycanbewrittenas 2TheUniformBoundednessPrincipleandtheClosedGraphTheorem,and/c)Corollary2.11.UnderthesameassumptionsasinTheorem2.10,thereexistsasuchthat(14)dist(x,G(x,G)(x,L)Proof.Given0,thereexistsuchthat(x,G)(x,L)Property(13)appliedtosaysthatthereexistsuchthatItfollowsthat(x,G(x,G)(x,L)C).Finally,weobtain(14)bylettingTheconverseofCorollary2.11isalsotrue:Ifaretwoclosedlinearsubspacessuchthat(14)holds,thenisclosed(seeExercise2.16).beasubspaceofaBanachspace.AsubspaceissaidtobeatopologicalcomplementorsimplyaWeshallalsosaythat.Ifthisholds,theneverymaybewrittenasItfollowsfromTheorem2.10thattheprojectionoperators\r\rlinearoperators.(Thatpropertycouldalsoserveasade“nitionofcomplementarysubspaces.)1.Everyadmitsacomplement.Indeed,letbeabasisof.Everymaybewrittenas(x).UsingHahn…Banach(analyticform)„ormorepreciselyCorol-lary1.2„eachcanbeextendedbyacontinuouslinearfunctional.Itiseasytocheckthatisacomplementof2.Every“nitecodimensionadmitsacomplement.Itsuf“cestochooseany“nite-dimensionalspacesuchthatisclosedsinceitis“nite-dimensional). 2.4ComplementarySubspaces.RightandLeftInvertibilityofLinearOperatorsHereisatypicalexampleofthiskindofsituation.Letbeasubspaceof.Thenf,xisclosedandofcodimension.Indeed,letbeabasisof.Thenthereexistsuchthati,j,...,p.[Considerthemapde“nedby\n(x)andnotethatissurjective;otherwise,therewouldexist„byHahn…Banach(secondgeometricform)„some\t=0suchthat\n(x)whichisabsurd].Itiseasytocheckthatthevectorsarelinearlyindependentandthatthespacegeneratedbythesisacomplementof.AnotherproofofthefactthatthecodimensionofequalsthedimensionofispresentedinChapter11(Proposition11.11).3.InaHilbertspaceeveryclosedsubspaceadmitsacomplement(seeSection5.2).Itisimportanttoknowthatsomeclosedsubspaces(eveninre”exiveBanachspaces)havecomplement.Infact,aremarkableresultofJ.LindenstraussandL.Tzafriri[1]assertsthatineveryBanachspacethatisnotisomorphictoaHilbertspace,thereexistclosedsubspacesanycomplement.(E,F)rightinverseisanoperator(F,E)leftinverseisanoperator(F,E)suchthatOurnextresultsprovidenecessaryandsuf“cientconditionsfortheexistenceofsuchinverses.Theorem2.12.Assumethat(E,F)surjective.Thefollowingpropertiesareequivalent:admitsarightinverse.)N(T)admitsacomplementinProof.(ii).Letbearightinverseof.Itiseasytosee(pleasecheck)thatR(S)S(F)isacomplementofN(T) 2TheUniformBoundednessPrincipleandtheClosedGraphTheorem(i).LetbeacomplementofN(T).Letbethe(continuous)projectionoperatorfrom.Given,wedenotebyanysolutionoftheequation.Setandnotethatisindependentofthechoiceof.Itiseasytocheckthat(F,E)andthatInviewofRemark8andTheorem2.12,itiseasytoconstructsurjectivewithoutarightinverse.Indeed,letbeaclosedsubspacewithoutcomplement,let,andletbethecanonicalprojectionfrom(forthede“nitionandpropertiesofthequotientspace,seeSection11.2).Theorem2.13.Assumethat(E,F)injective.Thefollowingpropertiesareequivalent:admitsaleftinverse.R(T)T(E)isclosedandadmitsacomplementinProof.(ii).ItiseasytocheckthatR(T)isclosedandthatN(S)isacomplementR(T))f=TSfTSf)(i).LetbeacontinuousprojectionoperatorfromR(T).Let;sinceR(T),thereexistsauniquesuchthat.Set.Itisclearthat;moreover,iscontinuousbyCorollary2.7.2.5OrthogonalityRevisitedTherearesomesimpleformulasgivingtheorthogonalexpressionofasumorofanProposition2.14.betwoclosedsubspacesin.Then GL=(G+L), (16) GL=(G+L). ProofofItisclearthat;indeed,iff,x0.Conversely,wehaveandthus(notethatif;similarly.ThereforeProofofUsethesameargumentasfortheproofof(16).Corollary2.15.betwoclosedsubspacesin.Then G+L,(18)(GL)= G+L.(19) 2.5OrthogonalityRevisitedProof.UsePropositions1.9and2.14.Hereisadeeperresult.Theorem2.16.betwoclosedsubspacesinaBanachspace.Thefollowingpropertiesareequivalent:isclosedinisclosedinProof.(c)followsfrom(19).(d)(b)isobvious.Weareleftwiththeimplications(a)(d)and(b)(d).Inviewof(18)itsuf“cestoprovethat.Given,considerthefunctionalde“nedasfollows.Forevery.Set\t(x)f,aClearly,isindependentofthedecompositionof,andislinear.Ontheotherhand,byTheorem2.10wemaychooseadecompositionofinsuchawaythat,andthus\t(x)byacontinuouslinearfunctionalde“nedonallof(seeCorollary1.2).So,wehave(a).WeknowbyCorollary2.11thatthereexistsaconstantsuchthat(20)dist(f,G(f,G(f,LOntheotherhand,wehave(f,Gf,x[UseTheorem1.12with\t(x)(x)f,x\f(x)(x),whereSimilarly,wehave 2TheUniformBoundednessPrincipleandtheClosedGraphTheorem(f,Lf,xandalso(by(17))(23)dist(f,G(f,(G f,xCombining(20),(21),(22),and(23)weobtain(24)sup f,xf,xf,xItfollowsfrom(24)that BG+GL1 CB Indeed,supposebycontradictionthatthereexistedsome G+Lwith\nx0\n1andx0/ .Thentherewouldbeaclosedhyperplanein .Thus,therewouldexistsomeandsomesuchthatTherefore,wewouldhavewhichcontradicts(24),and(25)isproved.Finally,considerthespacewiththenormnormx,y]\n=andthespace withthenormof.Themapde“nedbybyx,yislinearandcontinuous.From(25)weknowthat T(B UsingStep2fromtheproofofTheorem2.6(openmappingtheorem)wecon-cludethatT(B Itfollowsthatissurjectivefrom,i.e., G+L. 2.6AnIntroductiontoUnboundedLinearOperators.De“nitionoftheAdjoint2.6AnIntroductiontoUnboundedLinearOperators.De“nitionoftheAdjointbetwoBanachspaces.AnunboundedlinearoperatorisalinearmapD(A)de“nedonalinearsubspaceD(A)withvaluesin.ThesetD(A)iscalledtheOnesaysthatisbounded)ifD(A)andifthereisaconstant0suchthatThenormofaboundedoperatorisde“nedby (E,F)\t= \nu\n. Itmayofcoursehappenthatanunboundedlinearoperatorturnsouttobebounded.Thisterminologyisslightlyinconsistent,butitiscommonlyusedanddoesnotleadtoanyconfusion.Herearesomeimportantde“nitionsandfurthernotation: Graphof={[u,AuD(A) RangeofR(A)D(A) KernelofN(A)D(A) AmapissaidtobeisclosedinInordertoprovethatanoperatorisclosed,oneproceedsingeneralasfollows.TakeasequenceD(A)suchthat.Thenchecktwofacts:D(A)Notethatitdoessuf“cetoconsidersequencessuchthat0in(andtoprovethatisclosed,thenN(A)isclosed;however,R(A)neednotbeclosed..Inpractice,mostunboundedoperatorsareandaredenselyde“nedD(A)isdenseinDe“nitionoftheadjointD(A)beanunboundedlinearoperatorthatisdenselyde“ned.WeshallintroduceanunboundedoperatorD(Aasfollows.First,onede“nesitsdomain:D(A0suchthatv,Au\f|D(A) 2TheUniformBoundednessPrincipleandtheClosedGraphTheoremItisclearthatD(Aisalinearsubspaceof.Weshallnowde“ne.GivenD(A,considerthemapD(A)de“nedbyg(u)v,AuD(A).Wehaveg(u)D(A).ByHahn…Banach(analyticform;seeTheorem1.1)thereexistsalinearmapthatextendsandsuchthatf(u)Itfollowsthat.Notethattheextensionof,sinceD(A)TheunboundedlinearoperatorD(Aiscalledthe.Inbrief,thefundamentalrelationbetweenisgivenby v,Auv,uD(A),D(A ItisnotnecessarytoinvokeHahn…Banachtoextend.Itsuf“cestousetheclassicalextensionbycontinuity,whichappliessinceD(A)isdense,uniformlycontinuousonD(A),andiscomplete(see,e.g.,H.L.Royden[1](Proposition11inChapter7)orJ.Dugundji[1](Theorem5.2inChapterXIV).ItmayhappenthatD(Aisnotdensein(evenifisclosed);butthisisaratherpathologicalsituation(seeExercise2.22).ItisalwaystruethatifisclosedthenD(Aisdenseinfortheweak(F,F)de“nedinChapter3(seeProblem9).Inparticular,ifisre”exive,thenD(Aisdenseinfortheusual(norm)topology(seeTheorem3.24).isaboundedoperatorthenisalsoaboundedoperator(fromand,moreover, (E,F) Indeed,itisclearthatD(A.Fromthebasicrelation,wehavev,u\f|\nwhichimpliesthatandthusWealsohavev,Au\f|\n 2.6AnIntroductiontoUnboundedLinearOperators.De“nitionoftheAdjointwhichimplies(byCorollary1.4)thatandthusProposition2.17.D(A)beadenselyde“nedunboundedlinearoperator.Thenisclosed,i.e.,isclosedinProof.D(Abesuchthat.Onehastocheckthat(a)D(Aand(b)Wehave,AuD(A).Atthelimitweobtainv,Auf,uD(A).D(Av,Au\f|\nD(A))ThegraphsofarerelatedbyaverysimpleorthogonalityrelationConsidertheisomorphismde“nedbybyv,f=[Šf,vD(A)beadenselyde“nedunboundedlinearoperator.Then I[G(A). Indeed,letletv,f,thenthenv,ff,uv,AuD(A)Šf,uv,AuD(A)[Šf,vHerearesomestandardorthogonalityrelationsbetweenrangesandkernels:Corollary2.18.D(A)beanunboundedlinearoperatorthatisdenselyde“nedandclosed.ThenN(A)R(AN(AR(A)N(A) R(AN(A R(A).(iv)Proof.Notethat(iii)and(iv)followdirectlyfrom(i)and(ii)combinedwithPropo-sition1.9.Thereisasimpleanddirectproofof(i)and(ii)(seeExercise2.18).However,itisinstructivetorelatethesefactstoProposition2.14bythefollowingdevice.Considerthespace,sothat,andthesubspaces 2TheUniformBoundednessPrincipleandtheClosedGraphTheoremItisveryeasytocheckthatN(A)R(A)N(AR(AProofofBy(29)wehaveR(A(by(16))N(A)(by(26))ProofofBy(27)wehaveR(A)(by(17))N(A(by(28))Itmayhappen,evenifisaboundedlinearoperator,thatN(A)\t= R(A(seeExercise2.23).However,itisalwaystruethatN(A)istheclosureR(Afortheweak(E,E)(seeProblem9).Inparticular,ifre”exivethenN(A) R(A2.7ACharacterizationofOperatorswithClosedRange.ACharacterizationofSurjectiveOperatorsThemainresultconcerningoperatorswithclosedrangeisthefollowing.Theorem2.19.D(A)beanunboundedlinearoperatorthatisdenselyde“nedandclosed.Thefollowingpropertiesareequivalent:R(A)isclosed,R(Aisclosed,R(A)N(A(iv)R(AN(A)Proof.WiththesamenotationasintheproofofCorollary2.18,wehaveisclosedin(see(27)),isclosedin(see(29)),(see(27)and(28)),(iv)(see(26)and(29)).TheconclusionthenfollowsfromTheorem2.16. 2.7OperatorswithClosedRange.SurjectiveOperatorsD(A)beaclosedunboundedlinearoperator.ThenR(A)isclosedifandonlyifthereexistsaconstantsuchthat(u,N(A))D(A)seeExercise2.14.ThenextresultprovidesausefulcharacterizationofTheorem2.20.D(A)beanunboundedlinearoperatorthatisdenselyde“nedandclosed.Thefollowingpropertiesareequivalent:issurjective,i.e.,R(A)thereisaconstantsuchthatD(AN(AR(Aisclosed.Theimplication(b)(a)issometimesusefulinpracticetoestablishthatanoperatorissurjective.Oneproceedsasfollows.Assumingthat,onetriestoprovethatindependentof).Thisiscalledthemethodofaprioriestimates.Oneisnotconcernedwiththequestionwhethertheequationadmitsasolution;oneassumesthatisapriorigivenandonetriestoestimateitsnorm.Proof.(b).SetD(AByhomogeneityitsuf“cestoprovethatisbounded.Forthispurpose„inviewofCorollary2.5(uniformboundednessprinciple)„wehaveonlytoshowthatanythesetisbounded(in.Sinceissurjective,thereissomeD(A)suchthat.Foreverywehavev,fv,Auv,uandthusv,f\f|\n(c).Suppose.Using(b)withweseethatCauchy,sothat.Sinceisclosed(byProposition2.17),weconcludethat(a).SinceR(Aisclosed,weinferfromTheorem2.19thatR(A)N(AThereisadualŽstatement.Theorem2.21.D(A)beanunboundedlinearoperatorthatisdenselyde“nedandclosed.Thefollowingpropertiesareequivalent: 2TheUniformBoundednessPrincipleandtheClosedGraphTheoremissurjective,i.e.,R(AthereisaconstantsuchthatD(A),N(A)R(A)isclosed.Proof.ItissimilartotheproofofTheorem2.20andweshallleaveitasanexercise.Ifoneassumesthatthatdim,thenthefollowingareequivalent:surjectiveinjectivesurjectiveinjectivewhichisindeedaclassicalresultforlinearoperatorsin“nite-dimensionalspaces.ThereasonthattheseequivalencesholdisthatR(A)R(Aare“nite-dimensional(andthusclosed).InthegeneralcaseonehasonlytheimplicationssurjectiveinjectivesurjectiveinjectiveTheconversesfail,asmaybeseenfromthefollowingsimpleexample.Let;foreveryandset .Itiseasytoseethatisaboundedoperatorandthat)isinjectivebutsurjective;R(A)R(Aisdenseandnotclosed.CommentsonChapter2Onemaywritedownexplicitlysomesimpleclosedsubspaceswithoutcomplement.Forexampleisaclosedsubspaceofwithoutcomplement;see,e.g.,C.DeVito[1](thenotationisexplainedinSection11.3).ThereareotherexamplesinW.Rudin[1](asubspaceof),G.Köthe[1],andB.Beauzamy[1](asubspace\t=MostoftheresultsinChapter2extendtoFréchetspaces(locallyconvexspacesthataremetrizableandcomplete).Therearemanypossibleextensions;see,e.g.,H.Schaefer[1],J.Horváth[1],R.Edwards[1],F.Treves[1],[3],G.Köthe[1].Theseextensionsaremotivatedbythetheoryofdistributions(seeL.Schwartz[1]),inwhichmanyimportantspacesareBanachspaces.FortheapplicationstothetheoryofpartialdifferentialequationsthereadermayconsultL.Hörmander[1]orF.Treves[1],[2],[3].TherearevariousextensionsoftheresultsofSection2.5inT.Kato[1]. 2.7ExercisesforChapter2ExercisesforChapter2 2.1 ContinuityofconvexfunctionsbeaBanachspaceandletbeaconvexl.s.c.function.D(\t)1.Provethatthereexisttwoconstants0andsuchthat\t(x))Hint:Givenanappropriate0,considerthesets\t(x)2.ProvethatrR0suchthat\t(x\t(xr,iMoreprecisely,onemaychoosechooseMŠ\t(x RŠr. 2.2 beavectorspaceandletbeafunctionwiththefollowingthreeproperties:p(xp(x)p(y)x,y(ii)foreach“xedthefunction\rp(x)iscontinuousfrom(iii)wheneverasequencep(y0,thenp(y0foreveryAssumethatisasequenceinsuchthatp(x0andisaboundedsequencein.Provethat0andthatp(Hint:Given0considerthesetsp(xDeducethatifisasequenceinsuchthatp(x0forsomeisasequenceinsuchthat,thenp(p(x) 2.3 betwoBanachspacesandletbeasequencein(E,F)AssumethatforeveryconvergesastoalimitdenotedbyShowthatif,then 2.4 betwoBanachspacesandletbeabilinearform(i)foreach“xed,themap\ra(x,y)iscontinuous;(ii)foreach“xed,themap\ra(x,y)iscontinuous.Provethatthereexistsaconstant0suchthata(x,y) 2TheUniformBoundednessPrincipleandtheClosedGraphTheoremTheoremHint:IntroducealinearoperatorandprovethatisboundedwiththehelpofCorollary2.5.] 2.5 beaBanachspaceandletbeasequenceofpositivenumberssuchthatlim0.Further,letbeasequenceinsatisfyingtheproperty,C(x)suchthatC(x)Provethatisbounded.bounded.Hint:Introduce 2.6 Locallyboundednonlinearmonotoneoperators.beBanachspaceandletD(A)beanysubsetin.A(nonlinear)mapD(A)issaidtobeifitsatis“esAy,xx,yD(A).1.LetD(A).Provethatthereexisttwoconstants0andsuchthatD(A)..Hint:ArguebycontradictionandconstructasequenceD(A)suchthat.Choose0suchthatB(x,r)D(A).Usethemonotonicityofandat.ApplyExercise2.5.]2.ProvethesameconclusionforapointpointconvD(A)3.Extendtheconclusionofquestion1tothecaseof,i.e.,foreveryD(A)isanonemptysubsetof;themonotonicityisde“nedasfollows:g,xx,yD(A),Ax,Ay. 2.7 beagivensequenceofrealnumbersandlet1.Assumeforeveryelement(thespaceisde“nedinSection11.3).Provethat 2.8 beaBanachspaceandletbealinearoperatorsatisfyingTx,xProvethatisaboundedoperator.[Twomethodsarepossible:(i)UseExercise2.6or(ii)Applytheclosedgraph 2.9 beaBanachspaceandletbealinearoperatorsatisfyingTx,yTy,xx,y 2.7ExercisesforChapter2Provethatisaboundedoperator. 2.10 betwoBanachspacesandlet(E,F)besurjective.1.Letbeanysubsetof.ProvethatT(M)isclosediniffN(T)isclosed2.DeducethatifisaclosedvectorspaceinanddimN(T),thenT(M)isclosed. 2.11 beaBanachspace,,andlet(E,F)besurjective.Provethatthereexists(F,E)suchthat,i.e.,hasarightinverseofofHint:DonotapplyTheorem2.12;trytode“neexplicitlyusingthecanonicalbasisof 2.12 betwoBanachspaceswithnorms.Let(E,F)besuchthatR(T)isclosedanddimN(T).Letdenoteanothernormonthatisweakerthan,i.e.,ProvethatthereexistsaconstantsuchthatthatHint:Arguebycontradiction.] 2.13 betwoBanachspaces.Provethattheset(E,F)admitsaleftinverseisopenin(E,F).).Hint:Prove“rstthattheset(E,F)isbijectiveisopenin(E,F) 2.14 betwoBanachspaces1.Let(E,F).ProvethatR(T)isclosediffthereexistsaconstant(x,N(T))))Hint:UsethequotientspaceE/N(T);seeSection11.2.]2.LetD(A)beaclosedunboundedoperator.ProvethatR(A)isclosediffthereexistsaconstantsuchthat(u,N(A))D(A).(A).Hint:Considertheoperator,whereD(A)withthegraphnormand 2TheUniformBoundednessPrincipleandtheClosedGraphTheorem 2.15 ,andbethreeBanachspaces.Let,F)andlet,F)besuchthatR(TR(TR(TR(TProvethatR(TR(Tareclosed.closed.Hint:ApplyExercise2.10tothemapde“nedbyT(x 2.16 beaBanachspace.Letbetwoclosedsubspacesof.Assumethatthereexistsaconstantsuchthat(x,G(x,L),Provethatisclosed. 2.17 0,1])withitsusualnorm.ConsidertheoperatorD(A)de“nedbyD(A) 1.Checkthat D(A)2.Is3.ConsidertheoperatorD(B)de“nedbyD(B) dt.IsBclosed? 2.18 betwoBanachspacesandletD(A)beadenselyde“nedunboundedoperator.1.ProvethatN(AR(A)N(A)R(A2.AssumingthatisalsoclosedprovethatN(A)R(A[Tryto“nddirectargumentsanddonotrelyontheproofofCorollary2.18.Forquestion2arguebycontradiction:supposethereissomeR(Asuchthatthatu,0]/G(A)andapplyHahn…Banach.] 2.19 beaBanachspaceandletD(A)beadenselyde“nedunboundedoperator.1.AssumethatthereexistsaconstantsuchthatAu,uD(A).ProvethatN(A)N(A 2.7ExercisesforChapter22.Conversely,assumethatN(A)N(A.Also,assumethatisclosedandR(A)isclosed.Provethatthereexistsaconstantsuchthat(1)holds. 2.20 betwoBanachspaces.Let(E,F)andletD(A)beanunboundedoperatorthatisdenselyde“nedandclosed.ConsidertheD(B)de“nedbyD(B)D(A),B1.Provethatisclosed.2.ProvethatD(BD(A 2.21 beanin“nite-dimensionalBanachspace.Fixanelement\t=andadiscontinuouslinearfunctional(suchfunctionalsexist;seeExercise1.5).Considertheoperatorde“nedbyD(A)E,Axf(x)a.1.DetermineN(A)R(A).2.Is3.DetermineD(A4.DetermineN(AR(A5.CompareN(A)R(AaswellasN(AR(A)6.ComparewiththeresultsofExercise2.18. 2.22 ThepurposeofthisexerciseistoconstructanunboundedoperatorD(A)thatisdenselyde“ned,closed,andsuchthat D(A\t=,sothat.ConsidertheoperatorD(A)de“nedbyD(A)1.Checkthatisdenselyde“nedandclosed.2.DetermineD(A,and D(A 2.23 ,sothat.Considertheoperator(E,E)de“nedby foreveryN(T)N(T)R(T,and R(TComparewithCorollary2.18. 2TheUniformBoundednessPrincipleandtheClosedGraphTheorem 2.24 ,andbethreeBanachspaces.LetD(A)beadenselyde“nedunboundedoperator.Let(F,G)andconsidertheoperatorD(B)de“nedbyD(B)D(A)1.Determine2.Prove(byanexample)thatbeclosedevenifisclosed. 2.25 ,andbethreeBanachspaces.1.Let(E,F)(F,G).Provethat2.Assumethat(E,F)isbijective.Provethatisbijectiveandthat 2.26 betwoBanachspacesandlet(E,F).Letbeaconvexfunction.AssumethatthereexistssomeelementinR(T)is“niteandcontinuous.\t(x)\f(Tx),xProvethatforeveryN(TN(Tg). 2.27 betwoBanachspacesandlet(E,F).AssumethatR(T)“nitecodimension,i.e.,thereexistsa“nite-dimensionalsubspacesuchthatR(T)R(T)ProvethatR(T)isclosed.