66CHAPTERIVXiscalledboundedifforeachneighborhoodWof0thereexistsapositi - PDF document

66CHAPTERIVXiscalledboundedifforeachneighborhoodWof0thereexistsapositivescalarcsuchthatScW:THEOREM4.1.(CharacterizationofNormableSpaces)LetXbealocallyconvextopologicalvectorspace.ThenXisanormablevectorspaceifandonlyifthereexistsaboundedconvexneighborhoodof0.PROOF.IfXisanormabletopologicalvectorspace,letk
kbeanormonXthatdeterminesthetopology.ThenB1isclearlyaboundedconvexneighborhoodof0.Conversely,letUbeaboundedconvexneighborhoodof0inX:WemayassumethatUissymmetric,since,inanyevent,U\(�U)isalsoboundedandconvex.Letbetheseminorm(Minkowskifunctional)onXassociatedtoUasinTheorem3.6.Weshowrstthatisactuallyanorm.Thus,letx6=0begiven,andchooseaconvexneighborhoodVof0suchthatx=2V:Notethat,iftx2V;thenjtj1:Choosec&#x]TJ/;ñ 9;&#x.963;&#x Tf ;.8;' 0;&#x Td[;0sothatUcV;andnotethatiftx2U;thentx2cV;whencejtjc:Therefore,recallingthedenitionof(x);(x)=1 sup�t0;tx2Ut;weseethat(x)1=c�0;showingthatisanorm.Wemustshownallythatthegiventopologyagreeswiththeonedenedbythenorm:Since,byTheorem3.6,iscontinuous,itfol-lowsimmediatelythatB=�1(�1;)isopeninthegiventopol-ogy,showingthatthetopologydenedbythenormiscontainedinthegiventopology.Conversely,ifVisanopensubsetofthegiventopol-ogyandx2V;letWbeaneighborhoodof0suchthatx+WV:Choosec�0sothatUcW:AgainusingTheorem3.6,weseethatB1=�1(�1;1)UcW;whenceB1=c=�1(�1;(1=c))W;andx+B1=cV:ThisshowsthatVisopeninthetopologydenedbythenorm.Q.E.D.EXERCISE4.1.(a)(CharacterizationofBanachSpaces)LetXbeanormedlinearspace.ShowthatXisaBanachspaceifandonlyifeveryabsolutelysummableinniteseriesinXissummableinX:(AninniteseriesPxnisabsolutelysummableinXifPkxnk1:)HINT:IffyngisaCauchysequenceinX;chooseasubsequencefynkgforwhichkynk�ynk+1k2�k:(b)UsepartatoverifythatallthespacesLp(R),1p1;areBanachspaces,asisC0(
): 68CHAPTERIV(Ofcourse,gisnotanelementofS:)(c)LetnbetheintegerfrompartaandletfbeaC1functionwithcompactsupportsuchthatjf(x)j1forallxandf(0)=1:ForeachintegerM�0;denegM(x)=g(x)f(x�M);wheregisthefunctionfrompartb.ShowthateachgM2Sandthatthereexistsapositiveconstantcsuchthatn(gM)cforallM;i.e.,(=c)gM2VforallM:Further,showthatforeachM2;n+1(gM)p M:(d)ShowthattheneighborhoodVof0frompartaisnotboundedinS:HINT:DeneWtobetheneighborhood�1n+1(�1;1);andshowthatnomultipleofWcontainsV:(e)ConcludethatSisnotnormable.THEOREM4.2.(SubspacesandQuotientSpaces)LetXbeaBa-nachspaceandletMbeaclosedlinearsubspace.(1)MisaBanachspacewithrespecttotherestrictiontoMofthenormonX:(2)Ifx+MisacosetofM;andifkx+Mkisdenedbykx+Mk=infy2x+Mkyk=infm2Mkx+mk;thenthequotientspaceX=MisaBanachspacewithrespecttothisdenitionofnorm.(3)ThequotienttopologyonX=Magreeswiththetopologydeter-minedbythenormonX=Mdenedinpart2.PROOF.Miscertainlyanormedlinearspacewithrespecttotherestrictednorm.SinceitisaclosedsubspaceofthecompletemetricspaceX;itisitselfacompletemetricspace,andthisprovespart1.Weleaveittotheexercisethatfollowstoshowthatthegivendeni-tionofkx+MkdoesmakeX=Manormedlinearspace.Letusshowthatthismetricspaceiscomplete.Thusletfxn+MgbeaCauchysequenceinX=M:ItwillsucetoshowthatsomesubsequencehasalimitinX=M:WemayreplacethisCauchysequencebyasubsequenceforwhichk(xn+1+M)�(xn+M)k=k(xn+1�xn)+Mk2�(n+1):Then,wemaychooseelementsfyngofXsuchthatforeachn1wehaveyn2(xn+1�xn)+M; 70CHAPTERIVforallx2X:Deducethat,iftwonormsk
k1andk
k2determineidenticaltopologiesonavectorspaceX;thenthereexistconstantsC1andC2suchthatkxk1C1kxk2C2kxk1forallx2X:(b)SupposeSisalineartransformationofanormedlinearspaceXintoatopologicalvectorspaceY:AssumethatS( B1)containsaneighborhoodUof0inY:ProvethatSisanopenmapofXontoY:WecomenexttooneoftheimportantapplicationsoftheBairecat-egorytheoreminfunctionalanalysis.THEOREM4.3.(IsomorphismTheorem)SupposeSisacontinuouslinearisomorphismofaBanachspaceXontoaBanachspaceY:ThenS�1iscontinuous,andXandYaretopologicallyisomorphic.PROOF.Foreachpositiveintegern;letAnbetheclosureinYofS( Bn):Then,sinceSisonto,Y=[An:BecauseYisacompletemetricspace,itfollowsfromtheBairecategorytheoremthatsomeAn;sayAN;musthavenonemptyinterior.Therefore,lety02Yand�0besuchthatB(y0)AN:Letx02XbetheuniqueelementforwhichS(x0)=y0;andletkbeanintegerlargerthankx0k:ThenAN+kcontainsAN�y0;sothattheclosedsetAN+kcontains B(0):Thisimpliesthatifw2Ysatiseskwk;andifisanypositivenumber,thenthereexistsanx2XforwhichkS(x)�wkandkxkN+k:WriteM=(N+k)=:Itfollowsthenbyscalingthat,givenanyw2Yandany&#x-277;0;thereexistsanx2XsuchthatkS(x)�wkandkxkMkwk:Wewillusetheexistenceofsuchanxrecursivelybelow.WenowcompletetheproofbyshowingthatkS�1(w)k2Mkwkforallw2Y;whichwillimplythatS�1iscontinuous.Thus,letw2Ybegiven.Weconstructsequencesfxng;fwngandfngasfollows:Setw1=w;1=kwk=2;andchoosex1sothatkw1�S(x1)k1andkx1kMkw1k:Next,setw2=w1�S(x1);2=kwk=4;andchoosex2suchthatkw2�S(x2)k2andkx2kMkw2k(M=2)kwk:Continuinginductively,weconstructthesequencesfwng;fngandfxngsothatwn=wn�1�S(xn�1);n=kwk=2n; 72CHAPTERIVequippedwiththenormkfk=sup01jf(x)j:DeneT:X!YbyT(f)=f0:ProvethatXandYareBanachspacesandthatTisacontinuouslineartransformation.(b)NowletXbethevectorspaceofallabsolutelycontinuousfunc-tionsfon[0;1];forwhichf(0)=0andwhosederivativef0isinLp(forsomexed1p1).DeneanormonXbykfk=kfkp:LetY=Lp;anddeneT:X!YbyT(f)=f0:ProvethatTisnotcontinuous,butthatthegraphofTisclosedinXY:Howdoesthisexamplerelatetotheprecedingtheorem?(c)ProveanalogousresultstoTheorems4.3,4.4,and4.5forlocallyconvex,Frechetspaces.DEFINITION.LetXandYbenormedlinearspaces.ByL(X;Y)weshallmeanthesetofallcontinuouslineartransformationsfromXintoY:WerefertoelementsofL(X;Y)asoperatorsfromXtoY:IfT2L(X;Y);wedenethenormofT;denotedbykTk;bykTk=supkxk1kT(x)k:EXERCISE4.6.LetXandYbenormedlinearspaces.(a)LetTbealineartransformationofXintoY:VerifythatT2L(X;Y)ifandonlyifkTk=supkxk1kT(x)k1:(b)LetTbeinL(X;Y):ShowthatthenormofTistheinmumofallnumbersMforwhichkT(x)kMkxkforallx2X:(c)Foreachx2XandT2L(X;Y);showthatkT(x)kkTkkxk:THEOREM4.6.LetXandYbenormedlinearspaces.(1)ThesetL(X;Y)isavectorspacewithrespecttopointwiseaddi-tionandscalarmultiplication.IfXandYarecomplexnormedlinearspaces,thenL(X;Y)isacomplexvectorspace.(2)L(X;Y);equippedwiththenormdenedabove,isanormedlin-earspace.(3)IfYisaBanachspace,thenL(X;Y)isaBanachspace.PROOF.Weprovepart3andleaveparts1and2totheexercises.Thus,supposeYisaBanachspace,andletfTngbeaCauchysequenceinL(X;Y):ThenthesequencefkTnkgisbounded,andweletMbea 74CHAPTERIVNow,givenanonzerox2X;wewritez=(=2kxk)x:So,foranyn;kTn(x)k=(2kxk=)kTn(z)k(2kxk=)(2J)=Mkxk;whereM=4J=:ItfollowsthenthatkTnkMforalln;asdesired.THEOREM4.8.LetXbeaBanachspace,letYbeanormedlin-earspace,letfTngbeasequenceofelementsofL(X;Y);andsupposethatfTngconvergespointwisetoafunctionT:X!Y:ThenTisacontinuouslineartransformationofXintoY;i.e.,thepointwiselimitofasequenceofcontinuouslineartransformationsfromaBanachspaceintoanormedlinearspaceiscontinuousandlinear.PROOF.Itisimmediatethatthepointwiselimit(whenitexists)ofasequenceoflineartransformationsisagainlinear.SinceanyconvergentsequenceinY;e.g.,fTn(x)g;isbounded,itfollowsfromtheprecedingtheoremthatthereexistsanMsothatkTnkMforalln;whencekTn(x)kMkxkforallnandallx2X:Therefore,kT(x)kMkxkforallx;andthisimpliesthatTiscontinuous.EXERCISE4.8.(a)ExtendtheUniformBoundednessPrinciplefromasequencetoasetSofelementsofL(X;Y):(b)RestatetheUniformBoundednessPrincipleforasequenceffngofcontinuouslinearfunctionals,i.e.,forasequenceinL(X;R)orL(X;C):(c)Letccdenotethevectorspaceofallsequencesfajg;j=1;2;:::thatareeventually0,anddeneanormonccbykfajgk=maxjajj:Denealinearfunctionalfnonccbyfn(fajg)=nan:Provethatthese-quenceffngisasequenceofcontinuouslinearfunctionalsthatispoint-wiseboundedbutnotuniformlyboundedinnorm.Whydoesn'tthiscontradicttheUniformBoundednessPrinciple?(d)Letccbeasinpartc.Deneasequenceffngoflinearfunctionalsonccbyfn(fajg)=Pnj=1aj:Showthatffngisasequenceofcontinu-ouslinearfunctionalsthatconvergespointwisetoadiscontinuouslinearfunctional.Whydoesn'tthiscontradictTheorem4.8?(e)Letc0denotetheBanachspaceofsequencesa0;a1;:::forwhichliman=0;wherethenormonc0isgivenbykfangk=maxjanj: 76CHAPTERIVEXERCISE4.12.LetXbeanormedlinearspace,andlet XdenotethecompletionofthemetricspaceX(e.g.,thespaceofequivalenceclassesofCauchysequencesinX).Showthat XisinanaturalwayaBanachspacewithXisometricallyimbeddedasadensesubspace.THEOREM4.9.(Hahn-BanachTheorem,NormedLinearSpaceVersion)LetYbeasubspaceofanormedlinearspaceX:SupposefisacontinuouslinearfunctionalonY;i.e.,f2L(Y;R):ThenthereexistsacontinuouslinearfunctionalgonX;i.e.,anelementofL(X;R);suchthat(1)gisanextensionoff:(2)kgk=kfk:PROOF.IfisdenedonXby(x)=kfkkxk;thenisaseminormonX:Clearly,f(y)jf(y)jkfkkyk=(y)forally2Y:BytheseminormversionoftheHahn-BanachTheorem,thereexistsalinearfunctionalgonX;whichisanextensionoff;suchthatg(x)(x)=kfkkxk;forallx2X;andthisimpliesthatgiscontinuous,andkgkkfk:Obviouslykgkkfksincegisanextensionoff:EXERCISE4.13.(a)LetXbeanormedlinearspaceandletx2X:Showthatkxk=supff(x);wherethesupremumistakenoverallcontinuouslinearfunctionalsfforwhichkfk1:Show,infact,thatthissupremumisactuallyattained.(b)UsepartatoderivetheintegralformofMinkowski'sinequality.Thatis,if(X;)isa-nitemeasurespace,andF(x;y)isa-measurablefunctiononXX;then(ZjZF(x;y)dyjpdx)1=pZ(ZjF(x;y)jpdx)1=pdy;where1p1:(c)Let1p1;andletXbethecomplexBanachspaceLp(R):Letp0besuchthat1=p+1=p0=1;andletDbeadensesubspaceofLp0(R):Iff2X;showthatkfkp=supkgkp0=1jZf(x)g(x)dxj: 78CHAPTERIVkfkp=1:Ifthetheoremholdsforallsuchf;itwillholdforallf2D:(Why?)BecauseT(f)belongstoLq0andtoLq1byhypothesis,itfollowsthatT(f)2Lq;sothatitisonlytheinequalityonthenormsthatwemustverify.WewillshowthatjR[T(f)](y)g(y)dyjmp;wheneverg2D\Lq0withkgkq0=1:Thiswillcompletetheproof(seeExercise4.13).Thus,letgbesuchafunction.Writef=Pnj=1ajAjandg=Pmk=1bkBk;forfAjgandfBkgdisjointboundedmeasurablesetsandajandbknonzerocomplexnumbers.Foreachz2C;dene(z)=(1�z)=p0+z=p1and(z)=(1�z)=q00+z=q01:Notethat(t)=1=pand(t)=1=q0:Weextendthedenitionofthesignumfunctiontothecomplexplaneasfollows:Ifisanonzerocomplexnumber,denesgn()tobe=jj:Foreachcomplexz;denethesimplefunctionsfz=nXj=1sgn(aj)jajj(z)=(t)Ajandgz=mXk=1sgn(bk)jbkj(z)=(t)Bk;andnallyputF(z)=Z[T(fz)](y)gz(y)dy=nXj=1mXk=1sgn(aj)sgn(bk)jajjp(z)jbkjq0(z)Z[T(Aj)](y)Bk(y)dy=nXj=1mXk=1cjkedjkz;wherethecjk'sarecomplexnumbersandthedjk'sarerealnumbers.ObservethatFisanentirefunctionofthecomplexvariablez;andthatitisboundedontheclosedstrip0z1:Notealsothat 80CHAPTERIVforall0t1:PROOF.Wemayassumethatm0andm1arepositive.DeneafunctionGonthestrip0z1byG(z)=F(z)=m1�z0mz1:ThenGiscontinuousandboundedonthisstripandisanalyticontheopenstrip0z1:Itwillsucetoprovethatsups2RjG(t+is)j1:Foreachpositiveintegern;deneGn(z)=G(z)ez2=n:TheneachfunctionGniscontinuousandboundedonthestrip0z1andanalyticontheopenstrip0z1:Also,G(z)=limGn(z)forallzinthestrip.ItwillsucethentoshowthatlimjGn(z)j1foreachzforwhich0z1:Fixz0=x0+iy0intheopenstrip,andchooseaY&#x]TJ/;ñ 9;&#x.963;&#x Tf ;.4;! 0;&#x Td[;jy0jsuchthatjGn(z)j=jGn(x+iy)j=jG(z)je(x2�y2)=n1wheneverjyjY:Let�betherectangularcontourdeterminedbythefourpoints(0;�Y);(1;�Y);(1;Y);and(0;Y):Then,bytheMaximumModulusTheorem,wehavejGn(z0)jmaxz2�jGn(z)jmax(1;sups2RjGn(1+is)j;1;sups2RjGn(is)j)=e1=n;provingthatlimjGn(z0)j1;andthiscompletestheproofofthelemma.EXERCISE4.15.VerifythattheRieszInterpolationTheoremholdswithRreplacedbyanyregular-nitemeasurespace.