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1、90Chapter5BanachSpacesManylinearequationsmaybeformulatedintermsofasuitablelinearoperatoractingonaBanachspace.Inthischapter,westudyBanachspacesandlinearoper-atorsactingonBanachspacesingreaterdetail.WegivethedenitionofaBanachspaceandillustrateitwithanumberofexa
2、mples.Weshowthatalinearoperatoriscontinuousifandonlyifitisbounded,denethenormofaboundedlinearop-erator,andstudysomepropertiesofboundedlinearoperators.Unboundedlinearoperatorsarealsoimportantinapplications:forexample,dierentialoperatorsaretypicallyunbounded.W
3、ewillstudytheminlaterchapters,inthesimplercontextofHilbertspaces.5.1BanachspacesAnormedlinearspaceisametricspacewithrespecttothemetricdderivedfromitsnorm,whered(x;y)=kx yk.Denition5.1ABanachspaceisanormedlinearspacethatisacompletemetricspacewithrespecttotheme
4、tricderivedfromitsnorm.Thefollowingexamplesillustratethedenition.Wewillstudymanyoftheseexamplesingreaterdetaillateron,sowedonotpresentproofshere.Example5.2For1p<1,wedenethep-normonRn(orCn)byk(x;x;:::;x)k=(jxjp+jxjp+:::+jxjp)1=p:12np12nForp=1,wedenethe1,orm
5、aximum,normbyk(x1;x2;:::;xn)k1=maxfjx1j;jx2j;:::;jxnjg:ThenRnequippedwiththep-normisanite-dimensionalBanachspacefor1p1.9192BanachSpacesExample5.3ThespaceC([a;b])ofcontinuous,real-valued(orcomplex-valued)functionson[a;b]withthesup-normisaBanachspace.Moregene
6、rally,thespaceC(K)ofcontinuousfunctionsonacompactmetricspaceKequippedwiththesup-normisaBanachspace.Example5.4ThespaceCk([a;b])ofk-timescontinuouslydierentiablefunctionson[a;b]isnotaBanachspacewithrespecttothesup-normkk1fork1,sincetheuniformlimitofcontinuous
7、lydierentiablefunctionsneednotbedierentiable.WedenetheCk-normby0(k)kfkCk=kfk1+kfk1+:::+kfk1:ThenCk([a;b])isaBanachspacewithrespecttotheCk-norm.ConvergencewithrespecttotheCk-normisuniformconvergenceoffunctionsandtheirrstkderiva-tives.Example5.5For1p<1,thes
8、equencespace`p(N)consistsofallinnitesequencesx=(x)1suchthatnn=1X1pjxnj<1;n=1withthep-norm,!1=pX1kxk=jxjp:pnn=1Forp=1,thesequencespace`1(N)consistsofallboundedsequences,withkxk1=su