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首页 functional_analysis_notes(2011)

functional_analysis_notes(2011).pdf

functional_analysis_notes(2011)

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2016-12-19 0人阅读 举报 0 0 暂无简介

简介:本文档为《functional_analysis_notes(2011)pdf》,可适用于人文社科领域

FUNCTIONALANALYSISNOTES()MrAndrwPinchuckDpartmntofMathmatics(PurApplid)RhodsUnirsityContntsIntroductionLinarSpacsIntroductonSubstsofalinarspacSubspacsandConxStsQuotintSpacDirctSumsandProjcti<spanclass='wran'>ons<span>ThHoldrandMinkowskiInqualitisNormdLinarSpacsPrliminarisQuotintNormandQuotintMapCompltnssofNormdLinarSpacsSrisinNormdLinarSpacsBoundd,TotallyBoundd,andCompactSubstsofaNormdLinarSpacFinitDimnsionalNormdLinarSpacsSparablSpacsandSchaudrBassHilbrtSpacsIntroductionCompltnssofInnrProductSpacsOrthogonalityBstApproximationinHilbrtSpacsOrthonormalStsandOrthonormalBassBounddLinarOpratorsandFunctionalsIntroductionExamplsofDualSpacsThDualSpacofaHilbrtSpacThHahnBanachThormanditsC<spanclass='wran'>ons<span>quncsIntroductionC<spanclass='wran'>ons<span>quncsofthHahnBanachExtnsionThormBidualofanormdlinarspacandRflxiityThAdjointOpratorWakTopologisFUNCTIONALANALYSISALPBairrsquosCatgoryThormanditsApplicati<spanclass='wran'>ons<span>IntroductionUniformBounddnssPrinciplThOpnMappingThormClosdGraphThormFUNCTIONALANALYSISALPIntroductionThscoursnotsaradaptdfromthoriginalcoursnotswrittnbyProfSizwMabizlawhnhlastgathiscoursintowhomIamindbtdIthusmaknoclaimsoforiginalitybuthamadsralchangsthroughoutInparticular,Ihaattmptdtomotiatthsrsultsintrmsofapplicati<spanclass='wran'>ons<span>inscincandinothrimportantbranchsofmathmaticsFunctionalanalysisisthbranchofmathmatics,spcificallyofanalysis,concrndwiththstudyofctorspacsandopratorsactingonthmItisssntiallywhrlinaralgbramtsanalysisThatis,animportantpartoffunctionalanalysisisthstudyofctorspacsndowdwithtopologicalstructurFunctionalanalysisarosinthstudyoftansformati<spanclass='wran'>ons<span>offuncti<spanclass='wran'>ons<span>,suchasthFourirtransform,andinthstudyofdiffrntialandintgralquati<spanclass='wran'>ons<span>ThfoundingandarlydlopmntoffunctionalanalysisislarglydutoagroupofPolishmathmaticiansaroundStfanBanachinthfirsthalfofththcnturybutcontinustobanaraofintnsirsarchtothisdayFunctionalanalysishasitsmainapplicationsindiffrntialquati<spanclass='wran'>ons<span>,probabilitythory,quantummchanicsandmasurthoryamongstothrarasandcanbstbiwdasapowrfulcollctionoftoolsthathafarrachingc<spanclass='wran'>ons<span>quncsAsaprrquisitforthiscours,thradrmustbfamiliarwithlinaralgbrauptothllofastandardscondyarunirsitycoursandbfamiliarwithralanalysisThaimofthiscoursistointroducthstudnttothkyidasoffunctionalanalysisItshouldbrmmbrdhowrthatwonlyscratchthsurfacofthisastarainthiscoursWxaminnormdlinarspacs,Hilbrtspacs,bounddlinaroprators,dualspacsandthmostfamousandimportantrsultsinfunctionalanalysissuchasthHahnBanachthorm,Bairscatgorythorm,thuniformbounddnssprincipl,thopnmappingthormandthclosdgraphthormWattmpttogijustificati<spanclass='wran'>ons<span>andmotiati<spanclass='wran'>ons<span>forthidasdlopdaswgoalongThroughoutthnots,youwillnoticthatthrarxrcissanditisuptothstudnttoworkthroughthsIncrtaincass,thrarstatmntsmadwithoutjustificationandoncagainitisuptothstudnttorigourouslyrifythsrsultsForfurthrradingonthstopicsthradrisrfrrdtothfollowingtxts:�GBACHMAN,LNARICI,FunctionalAnalysis,AcadmicPrss,NY�EKREYSZIG,IntroductoryFunctionalAnalysis,JohnWilys<spanclass='wran'>ons<span>,NwYorkChichstrBrisbanToronto,�GFSIMMONS,Introductiontotopologyandmodr<spanclass='wran'>nana<span>lysis,McGrawHillBookCompany,Singapor,�AETAYLOR,IntroductiontoFunctionalAnalysis,JohnWilyS<spanclass='wran'>ons<span>,NYIhaalsofoundWikipdiatobquitusfulasagnralrfrncChaptrLinarSpacsIntroductonInthisfirstchaptrwriwthimportantnoti<spanclass='wran'>ons<span>associatdwithctorspacsWalsostatandprosomwllknowninqualitisthatwillhaimportantc<spanclass='wran'>ons<span>quncsinthfollowingchaptrUnlssothrwisstatd,wshalldnotbyRthfildofralnumbrsandbyCthfildofcomplxnumbrsLtFdnotithrRorCDfinitionAlinarspacorafildFisanonmptystXwithtwooprati<spanclass='wran'>ons<span>CWX�X!X(calldaddition)and�WF�X!X(calldmultiplication)satisfyingthfollowingproprtis:xCyXwhnrxyXxCyDyCxforallxyXThrxistsauniqulmntinX,dnotdby,suchthatxCDCxDxforallxXAssociatdwithachxXisauniqulmntinX,dnotdby�x,suchthatxC�xD�xCxDxCyCzDxCyCzforallxyzX˛�xXforallxXandforall˛F˛�xCyD˛�xC˛�yforallxyXandall˛F˛C�xD˛�xC�xforallxXandall˛F˛�xD˛��xforallxXandall˛F�xDxforallxXWmphasizthatalinarspacisaquadruplXFC�whrXisthundrlyingst,Fafild,Caddition,and�multiplicationWhnnoconfusioncanariswshallidntifythlinarspacXFC�withthundrlyingstXToshowthatXisalinarspac,itsufficstoshowthatitisclosdundradditionandscalarmultiplicationoprati<spanclass='wran'>ons<span>Oncthishasbnshown,itisasytoshowthatallthothraxiomsholdFUNCTIONALANALYSISALPDfinitionAral(rspcomplx)linarspacisalinarspacorthral(rspcomplx)fildAlinarspacisalsocalldactorspacanditslmntsarcalldctorsExamplsForafixdpositiintgrn,ltXDFnDfxDxx:::xnWxiFiD:::ngndashthstofallntuplsofralorcomplxnumbrsDfinthoprati<spanclass='wran'>ons<span>ofadditionandscalarmultiplicationpointwisasfollows:ForallxDxx:::xnyDyy:::yninFnand˛F,xCyDxCyxCy:::xnCyn˛�xD˛x˛x:::˛xn:ThnFnisalinarspacorFLtXDCOEligabDfxWOEligab!FjxiscontinuousgDfinthoprati<spanclass='wran'>ons<span>ofadditionandscalarmultiplicationpointwis:ForallxyXandall˛R,dfinxCytDxtCytand˛�xtD˛xt�foralltOEligab:ThnCOEligabisaralctorspacSquncSpacs:Informally,asquncinXisalistofnumbrsindxdbyNEquialntly,asquncinXisafunctionxWN!Xginbyn!xnDxnWshalldnotasquncxx:::byxDxx:::Dxn:ThsquncspacsLtsdnotthstofallsquncsxDxnofralorcomplxnumbrsDfinthoprati<spanclass='wran'>ons<span>ofadditionandscalarmultiplicationpointwis:ForallxDxx:::,yDyy:::sandall˛F,dfinxCyDxCyxCy:::˛�xD˛x˛x::::ThnsisalinarspacorFThsquncspacLtDNdnotthstofallbounddsquncsofralorcomplxnumbrsThatis,allsquncsxDxnsuchthatsupiNjxij:Dfinthoprati<spanclass='wran'>ons<span>ofadditionandscalarmultiplicationpointwisasinxampl()ThnisalinarspacorFThsquncspacpDpN�pLtpdnotthstofallsquncsxDxnofralorcomplxnumbrssatisfyingthconditionXiDjxijp:Dfinthoprati<spanclass='wran'>ons<span>ofadditionandscalarmultiplicationpointwis:ForallxDxn,yDyninpandall˛F,dfinxCyDxCyxCy:::˛�xD˛x˛x::::FUNCTIONALANALYSISALPThnpisalinarspacorFProofLtxDxx:::,yDyy:::pWmustshowthatxCypSinc,forachiN,jxiCyijp�OEligmaxfjxijjyijgp�pmaxfjxijpjyijpg�pjxijpCjyijpitfollowsthatXiDjxiCyijp�pXiDjxijpCXiDjyijp!:Thus,xCypAlso,ifxDxnpand˛F,thnXiDj˛xijpDj˛jpXiDjxijp:Thatis,˛�xpThsquncspaccDcNLtcdnotthstofallconrgntsquncsxDxnofralorcomplxnumbrsThatis,cisthstofallsquncsxDxnsuchthatlimn!xnxistsDfinthoprati<spanclass='wran'>ons<span>ofadditionandscalarmultiplicationpointwisasinxampl()ThncisalinarspacorFThsquncspaccDcNLtcdnotthstofallsquncsxDxnofralorcomplxnumbrswhichconrgtozroThatis,cisthspacofallsquncsxDxnsuchthatlimn!xnDDfinthoprati<spanclass='wran'>ons<span>ofadditionandscalarmultiplicationpointwisasinxampl()ThncisalinarspacorFThsquncspacDNLtdnotthstofallsquncsxDxnofralorcomplxnumbrssuchthatxiDforallbinitlymanyindicsiDfinthoprati<spanclass='wran'>ons<span>ofadditionandscalarmultiplicationpointwisasinxampl()ThnisalinarspacorFFUNCTIONALANALYSISALPSubstsofalinarspacLtXbalinarspacorFxXandAandBsubstsofXand�FWshalldnotbyxCAWDfxCaWaAgACBWDfaCbWaAbBg�AWDf�aWaAg:SubspacsandConxStsDfinitionAsubstMofalinarspacXiscalldalinarsubspacofXif(a)xCyMforallxyM,and(b)�xMforallxMandforall�FClarly,asubstMofalinarspacXisalinarsubspacifandonlyifMCM�Mand�M�Mforall�FExamplsErylinarspacXhasatlasttwodistinguishdsubspacs:MDfgandMDXThsarcalldthimproprsubspacsofXAllothrsubspacsofXarcalldthproprsubspacsLtXDRThnthnontriiallinarsubspacsofXarstraightlinsthroughthoriginMDfxDxx:::xnWxiRiD:::ngisasubspacofRnMDfxWOElig�!RxcontinuousandxDgisasubspacofCOElig�MDfxWOElig�!RxcontinuousandxDgisnotasubspacofCOElig�ShowthatcisasubspacofcDfinitionLtKbasubstofalinarspacXThlinarhullofK,dnotdbylinKorspanK,isthintrsctionofalllinarsubspacsofXthatcontainKThlinarhullofKisalsocalldthlinarsubspacofXspannd(orgnratd)byKItisasytochckthatthintrsctionofacollctionoflinarsubspacsofXisalinarsubspacofXItthrforfollowsthatthlinarhullofasubstKofalinarspacXisagainalinarsubspacofXInfact,thlinarhullofasubstKofalinarspacXisthsmallstlinarsubspacofXwhichcontainsKPropositionLtKbasubstofalinarspacXThnthlinarhullofKisthstofallfinitlinarcombinati<spanclass='wran'>ons<span>oflmntsofKThatis,linKD:nXjD�jxjjxx:::xnK��:::�nFnN=:FUNCTIONALANALYSISALPProofExrcis�DfinitionAsubstfxx:::xngofalinarspacXissaidtoblinarlyindpndntifthquation˛xC˛xC���C˛nxnDonlyhasthtriialsolution˛D˛D���D˛nDOthrwis,thstfxx:::xngislinarlydpndntAsubstKofalinarspacXissaidtoblinarlyindpndntifryfinitsubstfxx:::xngofKislinarlyindpndntDfinitionIffxx:::xngisalinarlyindpndntsubstofXandXDlinfxx:::xng,thnXissaidtohadimnsionnInthiscaswsaythatfxx:::xngisabasisforthlinarspacXIfalinarspacXdosnothaafinitbasis,wsaythatitisinfinitdimnsionalExamplsThspacRnhasdimnsionnItsstandardbasisisf:::ng,whr,forachjD:::n,jisanntuplofralnumbrswithinthjthpositionandzroslswhri,jD::::::whroccursinthjthpositionThspacPnofpolynomialsofdgratmostnhasdimnsionnCItsstandardbasisisftt:::tngThfuncti<spanclass='wran'>ons<span>pacCOEligabisinfinitdimnsionalThspacsp,with�p�,arinfinitdimnsionalDfinitionLtKbasubstofalinarspacXWsaythat(a)Kisconxif�xC��yKwhnrxyKand�OElig(b)Kisbalancdif�xKwhnrxKandj�j�(c)KisabsolutlyconxifKisconxandbalancdRmarkItisasytorifythatKisabsolutlyconxifandonlyif�xC�yKwhnrxyKandj�jCj�j�ErylinarsubspacisabsolutlyconxDfinitionLtSbasubstofthlinarspacXThconxhullofS,dnotdcoS,isthintrsctionofallconxstsinXwhichcontainSSincthintrsctionofconxstsisconx,itfollowsthatcoSisthsmallstconxstwhichcontainsSThfollowingrsultisanaltrnatcharactrizationofcoSFUNCTIONALANALYSISALPPropositionLtSbanonmptysubstofalinarspacXThncoSisthstofallconxcombinati<spanclass='wran'>ons<span>oflmntsofSThatis,coSD:nXjD�jxjjxx:::xnS�j�jD:::nnXjD�jDnN=:ProofLtCdnotthstofallconxcombinati<spanclass='wran'>ons<span>oflmntsofSThatis,CD:nXjD�jxjjxx:::xnS�j�jD:::nnXjD�jDnN=:LtxyCand���ThnxDnX�ixiyDmX�iyi,whr�i�i�,nX�iD,mX�iD,andxiyiSThus�xC��yDnX��ixiCmX���iyiisalinarcombinationoflmntsofS,withnonngaticofficints,suchthatnX��iCmX���iD�nX�iC��mX�iD�C��D:Thatis,�xC��yCandCisconxClarlyS�CHnccoS�CWnowprothinclusionC�coSNotthat,bydfinition,S�coSLtxxS,����and�C�DThn,byconxityofcoS,�xC�xcoSAssumthatn�X�ixicoSwhnrxx:::xn�S,�j�,jD:::n�andn�XjD�jDLtxx:::xnSand��:::�nbsuchthat�j�,jD:::nandnXjD�jDIfn�XjD�jD,thn�nDHncnX�jxjD�nxncoSAssumthatDn�XjD�jThn�j�foralljD:::n�andn�XjD�jDBythinductionassumption,n�XjD�jxjcoSHncnXjD�jxjDn�XjD�jxjAC�nxncoS:ThusC�coS�QuotintSpacLtMbalinarsubspacofalinarspacXorFForallxyX,dfinx�ymodMrdquox�yM:FUNCTIONALANALYSISALPItisasytorifythat�dfinsanquialncrlationonXForxX,dnotbyOEligxWDfyXWx�ymodMgDfyXWx�yMgDxCMthcostofxwithrspcttoMThquotintspacX=Mc<spanclass='wran'>ons<span>istsofallthquialncclasssOEligx,xXThquotintspacisalsocalldafactorspacPropositionLtMbalinarsubspacofalinarspacXorFForxyXand�F,dfinthoprati<spanclass='wran'>ons<span>OEligxCOEligyDOEligxCyand��OEligxDOElig��x:ThnX=Misalinarspacwithrspcttothsoprati<spanclass='wran'>ons<span>ProofExrcis�Notthatthlinaroprati<spanclass='wran'>ons<span>onX=Marquialntlyginby:ForallxyXand�F,xCMCyCMDxCyCMand�xCMD�xCM:DfinitionLtMbalinarsubspacofalinarspacXorFThcodimnsionofMinXisdfindasthdimnsionofthquotintspacX=MItisdnotdbycodimMDdimX=MClarly,ifXDM,thnX=MDfgandsocodimXDDirctSumsandProjcti<spanclass='wran'>ons<span>DfinitionLtMandNblinarsubspacsofalinarspacXorFWsaythatXisadirctsumofMandNifXDMCNandMNDfg:IfXisadirctsumofMandN,wwritXDM˚NInthiscas,wsaythatM(rspN)isanalgbraiccomplmntofN(rspM)PropositionLtMandNblinarsubspacsofalinarspacXorFIfXDM˚N,thnachxXhasauniqurprsntationofthformxDmCnforsommMandnNProofExrcis�LtMandNblinarsubspacsofalinarspacXorFsuchthatXDM˚NThncodimMDdimNAlso,sincXDM˚N,dimXDdimMCdimNHncdimXDdimMCcodimM:ItfollowsthatifdimX,thncodimMDdimX�dimMThopratorPWX!XiscalldanalgbraicprojctionifPislinar(i,P˛xCyD˛PxCPyforallxyXand˛F)andPDP,i,PisidmpotntFUNCTIONALANALYSISALP<spanclass='wran'>P<span>ropositionLtMandNblinarsubspacsofalinarspacXorFsuchthatXDM˚NDfinPWX!XbyPxDm,whrxDmCn,withmMandnNThnPisanalgbraicprojctionofXontoMalongNMororMDPXandNDI�PXDkrPConrsly,ifPWX!Xisanalgbraicprojction,thnXDM˚N,whrMDPXandNDI�PXDkrPProofLinarityofP:LtxDmCnandyDmCn,whrmmMandnnNFor˛F,P˛xCyDP˛mCmC˛nCnD˛mCmD˛PxCPy:IdmpotncyofP:SincmDmC,withmMandN,whathatPmDmandhncPxDPmDmDPxThatis,PDPFinally,nDx�mDI�PxHncNDI�PXAlso,PxDifandonlyifxN,i,krPDNConrsly,ltxXandstmDPxandnDI�PxThnxDmCn,whrmMandnNWshowthatthisrprsntationisuniquIndd,ifxDmCnwhrmMandnN,thnmDPuandnDI�PforsomuXSincPDP,itfollowsthatPmDmandPnDHncmDPxDPmCPnDPmDmSimilarlynDn�ThHoldrandMinkowskiInqualitisWnowturnourattntiontothrimportantinqualitisThfirsttwoarrquirdmainlytoproththirdwhichisrquirdforourdiscussionaboutnormdlinarspacsinthsubsquntchaptrDfinitionLtpandqbpositiralnumbrsIfpandpCqD,orifpDandqD,orifpDandqD,thnwsaythatpandqarconjugatxponntsLmma(YoungrsquosInquality)Ltpandqbconjugatxponnts,withpqand˛�Thn˛�˛ppCqq:ProofIfpDDq,thnthinqualityfollowsfromthfactthat˛��Noticalso,thatif˛DorD,thnthinqualityfollowstriiallyIfpD,thnc<spanclass='wran'>ons<span>idrthfunctionfWOElig!Rginbyf˛D˛ppCqq�˛forfixd:Thn,f˛D˛p��Dwhn˛p�DThatis,whn˛Dp�DqpWnowapplythsconddriatitsttothcriticalpoint˛Dqpf˛Dp�˛p�forall˛:FUNCTIONALANALYSISALPThus,whaaglobalminimumat˛Dqp:ItisasilyrifidthatDfqp�f˛D˛ppCqq�˛,˛�˛ppCqqforach˛OElig�Thorm(HoldrrsquosInqualityforsquncs)Ltxnpandynq,whrpand=pC=qDThnXkDjxkykj�XkDjxkjp!pXkDjykjq!q:ProofIfXkDjxkjpDorXkDjykjqD,thnthinqualityholdsAssumthatXkDjxkjpDandXkDjykjqDThnforkD:::,wha,byLmma,thatjxkj�PkDjxkjp�p�jykj�PkDjykjq�q�pjxkjpPkDjxkjpCqjykjqPkDjykjq:Hnc,PkDjxkykj�PkDjxkjp�p�PkDjykjq�q�pCqD:Thatis,XkDjxkykj�XkDjxkjp!pXkDjykjq!q:�Thorm(MinkowskirsquosInqualityforsquncs)LtpandxnandynsquncsinpThnXkDjxkCykjp!p�XkDjxkjp!pCXkDjykjp!p:FUNCTIONALANALYSISALPProofLtqDpp�IfXkDjxkCykjpD,thnthinqualityholdsWthrforassumthatXkDjxkCykjpDThnXkDjxkCykjpDXkDjxkCykjp�jxkCykj�XkDjxkCykjp�jxkjCXkDjxkCykjp�jykj�XkDjxkCykjp�q!qXkDjxkjp!pCXkDjykjp!pDXkDjxkCykjp!qXkDjxkjp!pCXkDjykjp!p:DiidingbothsidsbyXkDjxkCykjp!q,whaXkDjxkCykjp!pDXkDjxkCykjp!�q�XkDjxkjp!pCXkDjykjp!p:�ExrcisShowthatthstofalln�mralmatricsisarallinarspacShowthatasubstMofalinarspacXisalinarsubspacifandonlyif˛xCyMforallxyMandall˛FProPropositionProPropositionProPropositionShowthatcisalinarsubspacofthlinarspacWhichofthfollowingsubstsarlinarsubspacsofthlinarspacCOElig�(a)MDfxCOElig�Wx�Dxg(b)MDfxCOElig�WZ�xtdtDg(c)MDfxCOElig�Wjxt�xtj�jt�tjforallttOElig�gFUNCTIONALANALYSISALPShowthatiffM�gisafamilyoflinarsubspacsofalinarspacX,thnMD�M�isalinarsubspacofXIfMandNarlinarsubspacsofalinarspacX,undrwhatcondition(s)isMNalinarsubspacofXChaptrNormdLinarSpacsPrliminarisForustohaamaningfulnotionofconrgncitisncssaryforthLinarspactohaanotiondistancandthrforatopologydfindonitThisladsustothdfinitionofanormwhichinducsamtrictopologyi<spanclass='wran'>nana<span>turalwayDfinitionAnormonalinarspacXisaralaludfunctionk�kWX!Rwhichsatisfisthfollowingproprtis:ForallxyXand�F,Nkxk�NkxkDrdquoxDNk�xkDj�jkxkNkxCyk�kxkCkyk(TrianglInquality)AnormdlinarspacisapairXk�k,whrXisalinarspacandk�kanormonXThnumbrkxkiscalldthnormorlngthofxUnlssthrissomdangrofconfusion,wshallidntifythnormdlinarspacXk�kwiththundrlyinglinarspacXExampls(Examplsofnormdlinarspacs)LtXDFForachxX,dfinkxkDjxjThnXk�kisanormdlinarspacWgithproofforXDCProprtisNNarasytorifyWonlyrifyNLtxyCThnkxCykDjxCyjDxCyxCyDxCyxCyDxxCyxCxyCyyDjxjCxyCxyCjyjDjxjCRxyCjyj�jxjCjxyjCjyjDjxjCjxjjyjCjyjDjxjCjxjjyjCjyjDjxjCjyjDkxkCkyk:TakingthpositisquarrootbothsidsyildsN�FUNCTIONALANALYSISALPLtnbanaturalnumbrandXDFnForachxDxx:::xnX,dfinkxkpDnXiDjxijp!pfor�pandkxkDmax�i�njxij:ThnXk�kpandXk�karnormdlinarspacsWgiadtaildproofthatXk�kpisanormdlinarspacfor�pNForach�i�n,jxij�)nXiDjxijp�)nXiDjxijp!p�)kxkp�:NForanyxX,kxkpDrdquonXiDjxijp!pDrdquojxijpDforalliD:::nrdquoxiDforalliD:::nrdquoxD:NForanyxXandany�F,k�xkpDnXiDj�xijp!pDj�jpnXiDjxijp!pDj�jnXiDjxijp!pDj�jkxkp:NForanyxyX,kxCykpDnXiDjxiCyijp!p�nXiDjxijp!pCnXiDjyijp!pbyMinkowskisInqualityDkxkpCkykp:LtXDBOEligabbthstofallbounddralaludfuncti<spanclass='wran'>ons<span>onOEligabForachxX,dfinkxkDsupa�t�bjxtj:ThnXk�kisanormdlinarspacWprothtrianglinquality:ForanytOEligabandanyxyX,jxtCytj�jxtjCjytj�supa�t�bjxtjCsupa�t�bjytjDkxkCkyk:FUNCTIONALANALYSISALPSincthisistruforalltOEligab,whathatkxCykDsupa�t�bjxtCytj�kxkCkyk:LtXDCOEligabForachxX,dfinkxkDsupa�t�bjxtjkxkDbZajxtjdtA:ThnXk�kandXk�karnormdlinarspacsLtXDp�pForachxDxiX,dfinkxkpDXiNjxijp!p:ThnXk�kpisanormdlinarspacLtXDcorcForachxDxiX,dfinkxkDkxkDsupiNjxij:ThnXisanormdlinarspacLtXDLCnbthlinarspacofalln�ncomplxmatricsForALCn,lt�ADnXiDAiibthtracofAForALCn,dfinkAkDp�A�ADuutnXiDnXkDAkiAkiDuutnXiDnXkDjAkijwhrA�isthconjugattransposofthmatrixANotationLtabanlmntofanormdlinarspacXk�kandrBarDfxXjkx�akrgOpnballwithcntraandradiusrIBOEligarDfxXjkx�ak�rgClosdballwithcntraandradiusrISarDfxXjkx�akDrgSphrwithcntraandradiusr:FUNCTIONALANALYSISALP�yx�kxykD�yx�kxyk�yx�kxyk�EquialntNormsDfinitionLtk�kandk�kbtwodiffrntnormsdfindonthsamlinarspacXWsaythatk�kisquialnttok�kifthrarpositinumbrs˛andsuchthat˛kxk�kxk�kxkforallxX:ExamplLtXDFnForachxDxx:::xnX,ltkxkDnXiDjxijkxkDnXiDjxij!andkxkDmax�i�njxij:Whasnthatk�kk�kandk�karnormsonXWshowthatthsnormsarquialntEquialncofk�kandk�k:LtxDxx:::xnXForachkD:::n,jxkj�nXiDjxij)max�k�njxkj�nXiDjxijrdquokxk�kxk:Also,forkD:::n,jxkj�max�k�njxkjDkxk)nXiDjxij�nXiDkxkDnkxkrdquokxk�nkxk:Hnc,kxk�kxk�nkxkWnowshowthatk�kisquialnttok�kLtxDxx:::xnXForachkD:::n,jxkj�kxk)jxkj�kxk)nXiDjxij�nXiDkxkDnkxkrdquokxk�pnkxk:FUNCTIONALANALYSISALPAlso,forachkD:::n,jxkj�nXiDjxij!=Dkxk)max�k�njxkj�kxkrdquokxk�kxk:C<spanclass='wran'>ons<span>quntly,kxk�kxk�pnkxk,whichprosquialncofthnormsk�kandk�kItis,ofcours,obiousnowthatallththrnormsarquialnttoachothrWshallslatrthatallnormsonafinitdimnsionalnormdlinarspacarquialntExrcisLtNXdnotthstofnormsonalinarspacXFork�kandk�kinNX,dfinarlation#byk�k#k�kifandonlyifk�kisquialnttok�k:Showthat#isanquialncrlationonNX,i,#isrflxi,symmtric,andtransitiOpnandClosdStsDfinitionAsubstSofanormdlinarspacXk�kisopnifforachsSthrisan�suchthatBs��SAsubstFofanormdlinarspacXk�kisclosdifitscomplmntXnFisopnDfinitionLtSbasubstofanormdlinarspacXk�kWdfinthclosurofS,dnotdbyS,tobthintrsctionofallclosdstscontainingSItisasytoshowthatSisclosdifandonlyifSDSRcallthatamtriconastXisaralaludfunctiondWX�X!Rwhichsatisfisthfollowingproprtis:ForallxyzX,Mdxy�MdxyDrdquoxDyMdxyDdyxMdxz�dxyCdyzThorm(a)IfXk�kisanormdlinarspac,thndxyDkx�ykdfinsamtriconXSuchamtricdissaidtobinducdorgnratdbythnormk�kThus,rynormdlinarspacisamtricspac,andunlssothrwisspcifid,wshallhncforthrgardanynormdlinarspacasamtricspacwithrspcttothmtricinducdbyitsnorm(b)IfdisamtriconalinarspacXsatisfyingthproprtis:ForallxyzXandforall�F,idxyDdxCzyCz(TranslationInarianc)iid�x�yDj�jdxy(AbsolutHomognity)thnkxkDdxdfinsanormonXFUNCTIONALANALYSISALPPr

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