Real and Functional Analysis PDF
Preview Real and Functional Analysis
• • Serge lang Reala nd Functional Analysis ThirEdd ition Springer GraduaTteex tisnM athemat1ic4s 2 EditBoorairadl AxelrF . GehrPiR.n.Hg a lmos S. W. Sprienr g NewY ork Berlin Heidelberg Barcelona Budapest HonKgo ng London Milan Paris SanCtlaa ra Singapore Tokyo BooKosFR ELATIENTDE REBSYST E RGE LANG FundamentoalDiso phantine Geometry f A systemaatcicco unotf fundamentailnsc,l udtihnegb asitch eoroyf heights, Rotha ndS iegetlh'eso remtsh,eN eronT-ateq uadraftoircm t,h eM ordeii-Weill theoreWme,i la ndN eronfu nctionasn,dt hec anonifcoarlm o n a curvaes i t relatteotd h eJa cobivainat het hetfuan ction. IntroducttoiC oonm pleHxy perboSlpiacc es Sinciet si ntroductbiyo Kno bayashtihe,t heoroyf complehxy perboslpiacc es hasp rogrescsoends iderTahbilsyb .o okg ives aacnc ouonfts omeo ft hem ost importanrte sulstusc,h a s Brody'tsh eorehmy,p erboilmibce ddincgusr,v ature propertiaensd,s omeN evanlitnhneao rIyt.a lsion cludCeasn an'psr ooffo rt he SeconMda inT heorewmh,i cwha se legaanntds hort. ElliptCiucrv esDi:o phantiAnnea lysis Thiss ystemaatcicco uonftt he badsiiocp hantthineeo royn ellipctuircv esst arts with the cWleaisesrisctarplaa srsa metrizcaotmipolne,m enbtyet dh eb asitch eory ofN eronfu nctionasn,d g oeso nt ot hef ormaglr ouph,e ighatnsd t heM ordeii Weilt heoreamn,d b oundfso ri ntegproailn tAs s. econpda ng iveasn e xtensive accounotf Baker'mse thoidn d iophantaipnper oximaatnidod ni ophantiinne equaliwthiiecshw erea ppliteodg et tbhoeu ndfso rt he integrailn t phoei nts firpsatn . CyclotoFmiiecl Ida sn dI I Thisv olumper ovidaensu p-to-diantter oducttoti hoent heoroyfa concreatned classicvaelrlyiy n teresetxianmgp loef n umbefri eldIsti. s o fs peciianlt erteos t numbetrh eoriasltgse,b rgaeiocm etetrosp,o logainstdas l,g ebrawihsot wso rk in 5)9 K-theoTrhyi.s b ooki sa combineeddi tioofn (GTM and CyclotoFmiiecl ds (GTM6 9)w hicahr eo uto f prIinna td.d ittioos no mem inor CyclotoFmiieclI dsI correcttihoinessd ,i ticoonn taainn sa ppebnydK iaxr Rlu bipnr ovitnhgeM azurW-iles theor(etmh 'e' macionn ject)ui rnae s elf-conwtaayi.n ed • ' OTHEBRo oKsL ABNYPG u BLISBHYE D SPRINVGEERRL-AG IntroducttoAi roank elTohve oryR iemann-RAolcghe b(rwai th WilFluilatmo n) • • CompleMxu ltiptliiocnIa ntroducttoM ioodnu lFaorr msM odulUanri t(sw itDha niel • • Kuben)I ntroducttoAi logenb raaincd A beliFaunn ctioCnysc lotoFmiiecl Ida sn dI I • • EllipFtuincc tioNnusm beTrh eoryA lgebraic NumberS LT2h(eRo)Ar by elian • • • • • VarietiDeisf fereMnatniiaflo lCdosm pleAxn alysiRse aAln alysUinsd ergraduate • • • • AnalysiUsn dergradAulaegtber aL ineAalrg ebrIan troducttoLi ionne Aalrg ebra • • • • CalculoufsS everVaalr iableFsi rsCto ursien C alculuBsa siMca thematics • • • Geometr(yw:i tGhe neM urrow)M athE!n countweirtshH igh SchSotould ents • TheB eautyo fD oingM atehmaticsT HE FILE • • SergLea ng Reaaln d FunctioAnnaall ysis ThirEdd ition Wit37hI l lustrations Springer SergLea ng of DepartmentM athemiacts YalUen iversity 06520 CT New Haven, USA EditBoorairadl S.A xler F.WGe.h ring Halmos P.R. Departmeonft Departmeonft Departmeonft Mathematics Mathematics Mathematics MichgianS tatUen iversitUyn iversoiftM yi chigan SantCal aar University 48824 48109 95053 EasLta nsinMgI, AnnA rborM,I SantCal arCaA, USA USA USA MSC 1991:S ubjecCtl aisfiscation: 26-0218,- 04l6,- 01 Libraryo fC ongreCsast aloging-in-DPautbal ication LangS,e rg1e9, 27- Reaaln dfu nctioannaall ysSiesr gLea ng-. 3rde d. I p. em. -(Gradutaetxeit nsm atheamti;c1 s4 2) Inclubdiebsl iograrpehfiecraealnn cdei sn dex. ISBN0 -387-94001-4 Matheamticaanla lysIi.sT. i tlIeI.S. e ries. I. QA300.L2741 993 92-21208 515-dc20 CIP Thep reivouesd itiwoansp ublhiesda s Copyrig1h9t8 3 Addison-Weeys. l by ReaAln alysis. Printoenda ci-dfrepea per. © 199S3p rinrg-eVerlNaegw YorIkn,c . Allr igthsr esveerdT.h isw orkm ayn otb et ransloartc eodp ieidnw holoer i np arwti thout thew rittpeenr misisono ft hep ublish(eSrp rgienr-VleargN ew York, InFci.f,tA hv eune, 175 New YorkN,Y 10010,U SA) excfeoprbt r ieefxc erpitnsc onnectwiiotnrh e vieowrss chol arlayn alysUisse.i nc onnectni woitahn yf ormo fi nfoartmiosnt oargea ndr etrieevlaelc, tronaidca ptiaontc,om putesro fwtareo,r bsyi miloarrd issimmieltahro dolnoogwyk nown orh ereaftedre velopiesfd o rbdiedn. Theu seo fg enreadle sctriipvnea mest,r adnea mse,t rdaemarkest,c i.n,t hipsu blicateiveonn , ift hef ormearr en ote speciiadlelnyt ifiisen do,tt ob et aekna sa sigtnh astu chn amesa,s understoodb yt heT radMea rksa ndM erchadnisMea rksA ct,m ay accordinbgelu yse d freeblyya nynoe. Producticooonr dinabtyeB dr iaHno wea ndm anagebdyT erryK ornakm;a nfuacrtiunsgu pervibys ed VinceSncte l.t a Typesebty A scoT radTey pesetLttidn.Ng,o rth Poi,nH tongK ong. Printaenddb ounbdy R .R .D onnelleSyo nsH,a rirsongbu,Vr A. & Printientd h eU nitSetda toefsA m erica. 9 4 3 8 7 6 25 0-387-94001-4 ISBN SpirngeVre-rlag YNoerwk B erlHieni dleberg 3-540-94001-4 10504356 ISBN SpringVeerr-laBge rlHieni delbNeerwg Y ork SPIN Foreword Thbioso ikms e aansatt efxrot fi ar ysetga rra dcuoasuteire na nalysis. Any sdtc aonsudeira nur ndgerradaunaatlewy iscliolsn sets iutffiuctient prepafrroai ttuisno dne rs,ft roai nnadsnitcmneygU, n dergrAandaulate ysiIas .s sutmhea tr etaihdaseec rq ueadwi innttoht ioofun nosimr f con vergaenntdch leei ke. Int htihsie rddi tIih oanvre,e orgtahnbeio zoekdc obvye irnitneg gratbieoofrnfenu ctiaonnaallSy usciahsr .e arermaefinntgtts h wea y courasrteeas u ignh ta ll Ikt nhoeow f .ph laavaIced e dasen dmu beorf examapnledex csei rsaesws e,al sl smoamteea rbioiaunltt egornta htei on relail(n eeo. ng .D sierqaucae pnpcreo xainmodan Ft oiuornai nearl ysis), ansdo mmea teornfi nuaclt iaonnaal(lye st.ihgtse.h eooftr hyGe e alnfd trsaofnrimnC hapXtVeIrT) h.e uspeg rpardeeve ixoceuirsst eoss e ctions inth tee xt. Ina s entshese,u bjmeactctto evrte hrsesa mteo paisec lse mentary calcvuillznui.es a,ar l gedbierffrae,n tainaidten igtoronan tT.ih itsi me, howetvheesrsu,jeb e catrtser eaitname adn nseuri tfraot bhlteer aining ofp reosfsiio.npeae.lo swp,hl eow iulslte h teo oilnfsr u thienervs tiga tiobneis it,nm athateimocrps hs,yi ocrws ,h ahta ve you. In fitrhpseat r wteb, e gwiintp ho int seets steofnropta oillalol g y, analaynswdie s ,c overi pmtohretr aemnsotus ltt s. Ia ms elehceetr,si ivnetc hepi asri tsr egaarsda te odo els,p ecially ChapIta enrIdsI . rMeuaslnatyrs ee a sayn,ad r lee essss etnhtaina l thoisnte ht ee xTth.eh yae vb eegni vienen x erwchiisacerhdsee , s igned toa cqufciairleii nrt oyu ttienciehqe nusa ntdo flgeixvief broit lhiotsye whwoa nttoc overo fts hoeammteg relaetenrgT thpheo. i snettto pol ogsyi mdpelaywl istt hh e basicc oinnnitoyutt,oi poeannns cd l oofs ed sectosnn,e cnteescdso,m pacatnncdeo snst,if nnuucotuiTsoh ncesh .a pter . FOREWORD VI concecrnonitinnugfon uucst iooncn osm psaecpttrs o peemprhlsayi zes reswuhlcithas rl eamdiyax n alaynsudin osir fmc ovnergweintche the langoufpa ongitse e tto pology. Int hsee cpoanrdCt h,a pItVea rnsVd, w ed escbrriibteehfl ey two baslniiecsa pra caensa lonyfas mieBsla,yn a scpchaea sn Hdli besrpta ces. Thnee pxatdr eta eltxse nswiiviteenhlgt yr ation. Web egwiintt hhde e veloopftm hieenn tte grfasalh.ih oanTs h eb een toe mpahsipzoes iatniodvr idteypr rionpge( ritnicerasen addse icnrge as insge queInfi cnetdssh e) ix.c esTshtierv eea.gt vimeehnne tar tete pmts 1 tog iavp er obpaelra bnectewL e- ecno nvearngpdeo tnsictiveyFi .or mordee taciolmemdes nettehsi et,n r uotcditooPn a Trhtr eeC haanpdt er VI. Thceh aeprotsna ppliocfai tnitoengasrn addsti tiorni bpurvtdoiieo ns concerxeatmeap dnlc ehso ifrocl eedsai ntghce or usieno thdeierrc tions, att htea sotfe l tcehteeur rT.h gee netrhaeloof rit yne griantm ieaon suerds pa(cweisrt ehs pteoc gti vpeons imteiaevs)eau lrterwniatthe s a chpatseg rivsipnegcr iefisocuf ilt ntestg irooanne uclsipdaeocaren s t he relailn Neei.t hiessrl t iegadht etxhpeeeo nfts hoeht e.rI nt htihsdi r editIhi aovnae,d d seodm mea tiearlf nuocnnt sio ofb ounvdaerdi ation, anIdh aveem pahsicznoevdot liuoannsd atphper oxbiymD aitriaocn sequeonrfc maeisle iveemsno rtehn a tihnp eer vieoduist firooi nns, stainnc eC,h VaIpI§tI2e,.r Fowra notfa b etptleartc hece,a lucs(u wlivtahl iunea s B anach spancoeow)c causar s se papraaartftteed re alwiintitghne tgiroaann,d beofrteh fenu cotniaanla ilsy.s Thdeie ffrenctacilualiluds so nbee caautbs eesm to,sp te owpillol en ly bea cqtueawdii nitothn lyec uliinds epaaancdne i ,n coemlpayltt e htat. Moeri mportthacena tllciyunB,l a unsas cphah caeasscu qeidrc onsider abilmep oritnta hnleca ets wtdo e cabdeecsao,uf ms aen aypp lications liMkoer tsheye ,to hrcea lcoufvl aursi aatdnit ohNneas s-,hM oesri m plimcaiptpt ihonerge wmh,il ciehe vsfe runt hientrh d iiescro tni soinnec e hatsod ewailmt ohr e gsepnaetcrheaaBslna nascpha cesr.e tsTsuh lese perttoati hgnee omoefftn urtcyi sopna cCef.ts h.ee x erocfCi hsaepst er XIfVro s miplearp plications. Then expta rdte awlistf nhuc tiaonnaallTy hspei uspr.o sheee ri s twfolodW.e p latchee allignaeei banarr ni nifietdn imeanlss eitotning whecrnoet inausisptutymia ornmesa doent hlneie maarp asn,wd e s how howo ncea unl irniezaae p"r obblyet ma kdienrgvi evasag,tna i iina setwtheirntegeh t hoery bceaa pnip eldf nutcioto nc essTp.ha ipsa rt incelssu edavlme ajrosrp ectthoreraeolmf s yas,nsi ahslowhionwwge c an etxentdot hien fiend iitmencsaiscoeen ratlrsae utilson f fi nidtiem en sinoallei anarl gae.Tb hrceo mpaancdt d hForolepme rhavateoa rplspi cattioot nesig norpaelrs aa ntpdoa rrdtieiffraeln etlilaoilpp etri(acet .ogr.s inp apoefAri tsy anhg-aeSnriAd it yah-Bott). ChapXtIeaXrn sXd X IoXn.u n both�enirrlimteaodnp eractoomrbsi,n e .. FOREWORD VII botthhl ei naelagrea bnridatn eagtritohne ionr ys tthueod fsy u ch operaOtnomera svy.i etwh e tresapteamclmet enrats auosprof rev si ding ane xpalmoefg eneirtnaealgit rotnh ye oonrl ocacolmlpysa pcatc es, wheebraym euaersi osb tafirmno efadnu ctioontn hael scpoanctei nof uoufnusc twiioctnohsm psaucoptrp t. Ifi nidat p opprritaoit net rsotduudcteeodn ietffrse ntmiaanobillfdes duirntgh fiirss t gryaedaaurna atlecy orsusineso, ot n lbyeu csae these ojbecaetro sfi ntetrode ieseffrttn igaelo metdeiereffrsnl tt oiorap ologists, bubte cagulsoeb alo nam naanolilyhfdsascis os m ieon i tsto wnb,o tihn ititsne garnadle erdtniiffaaslp eIcittst sh e.eof errd esitroia enbgtlrea te manoilifdnas n alcyossueirassn, Id h avdeo ntehi ints h lea stw hpiacrht , maayl bes ov iaespw redo vigdoioandpg p liocfia entgtiroatnth ieoonr y. a nmubeorfe xamaprlgeei sv etnh texie tn mbauntiy n tteirnegs A examaprlaeel gssi ov ientn he exc eir(soefris nt sanecxep,fl roimcuifltroa s approxiwmhaoetsxietio esnnoscn eke n oawbssc ttbralyyt hWee ierstrass Stotneheoe rmi;tn egorpaelr atvoarrsik oidoun;sefs t cT)h.ee x ercises shobuelv diw eed anai stn egpraarlt of the Cbhoaeoprkts. Note that XIaXn d gXiXv,ti hnsegp ecmteraeas,lcu arn vibeeaw sep dr oviding ane xalmefpr om annyo tiwohnishc ahvb ee edni ssscepudr ev:i ously operaitnHo irlsbs epramtce ea,es su,r caonndv oluttihsoean mse. At timteh,er seuetsl sl eaddi reicont ttlhryee aanla loyfs iwtsoh rek ing matmhaetician. Asu sluI,ah avaev odia dsfer aa sp ossbiubillledo incnghg a oifn s lgoicitanelpr ednedeae,nn hdca evm adceh taeparsls o giicnadleelpnyet n d asp oses,si otb hlcaotu rwshheir scu rna pitdholruygc he rcthataiepnr s, omtiitnsgo mmeae tiral,c ocvlaeanrtc ehra pwtietrhbsoe uiltno gg ically inceoinnevnced. Thep resbeonoctka bne u esdf ro taw o-secmoeusrotsmeeiirg,nt t sommeae triIah lo.pI e hgaivvaees nua ibtolvee rovfi ebtwah steio co ls ofa nalTyheserim si.tg hsboem ree atsoioc nnl uodhteet ro psiucacssh, thbea stihce ocroenmncsie nergl loippetriacIt h oarvosem.t ei tdt his topaincd sootmheep rasrb,te lcya tuhsaeep peenstd omi ycS L(2R ) cnosuttiaets sbu -bowohkic cohn ttahietnsosep aincpdsa ,r btelcya use theirnseo t itmoe ctohveiemntr h bea soinycee caorsu ear ddressed to gruaadsttdeue nts. The pbroeosckea annlt bs eou seadsa r eerfefnrocb ea sainca lysis, sincoeeffr tsih rtee eartd hoep porttosu enlivetaicyrot tu osp wiictsh out readitnhegen etb ioroTkh.se buj emcatet rito sr geadn itszhoiam tta kes the atvoapbiittlceoaas s waind ea uadspi oesnesc.ie b l Theea rerm any gvoeobrdoy o kisen rtimne dainaatley siiensrt, and estriensge arcwhh ipccaahpn e rrebsaie,dm maedtiealfytt ehprer eenst courAsp ea.rl tiiisagstli vientn hB ei blioIgnrc attfp,ahdh eey t.e rmina tioofn mtahtee irnlicuadlie nd Rtehails F uanncdt Ainoanlahylas iss beegnrt elamyo taitvebdyt heex einscoteft hepsaep aenrbdso oks, and byt hnee etdop rovindeec eytbs hasecao krugnfrrdot hem. FOREWORD VIII FianlIlt yh,a anltklh ose wpheohvo aepm laedv ea lucaobmlmtees n ancdo riroe,nec sstpecKieaiCltolhrny a Mda,r tMionh le,nT kaakmepsi Yamaknaaa,n Sdt epChheinar piwp,ha roe vitehwbeeo dof rkoS rpniegr Verlag. NewH av1e99n3 9/916 SERLGAEN G Co nten st PART ONE GeneraTlo polog.y. . .. .. .. .. . .. . . . .. .. .. .. . 1. . . . . . . . . . . . . . . . . . . . . CHAPTEIR . . . .. .. .. .. .. .. .. .. .. . . . . .3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sets §l. . . . . . . . . . . . .3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . SomBea csT iermoignyo l §2. . . . . . . . . . . . . .7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DenumeSreatbsl e §3. . . . . . . . . . . . . . .1 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ZorLne'mmsa CHAPTEIRI . .. .. . .. . .. . .. . . . . . . .1 7 . . . . . . . . . . . . . . . . . . . . . . . . . . TopologiScpaalc es §l .O peann Cdl osed. .S e.t s. . . . . . .. .. . ..1 7. . . . . . . . . . . . . . . . . . . . . . . . . . §2C.o nneScett.es d . . . . . . . .. .. .. .. .. 2..7 .. .. .. .. .. .. .. .. .. .. . . . . . . . . §3C.o mpSaapccte .s . . . . . . .. . .. .. .. .. .. .. 3..1 .. .. .. .. .. .. .. .. . . . . . . . §4S.e parbayCt oinnotouniu Fsu nct.i o.n s. . .. .. .. .. .4. 0 . . .. .. .. .. .. . . . §5E.x erc. i.s e.s . . . . . . . . . ... .... . .4 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTEIRl l . . . . . . . . .5 1 . . . . . . . . . . . . . . . . . . . ContinuoFuusn ctioonns C ompactS ets . §l .T hSet oWneiee-rsTthreeaomsr. s . . . . . . .. .. .5. 1 . . . . . . . . . . . . . . . . . . . . §2I.d eoafCl onsit nusFo uunct.i o.n s. .. . ... . .. .. .. ... ... 5 ..5. . . . . . . . . . §3. . . . . . . . . . .. .. .. .. 5 .7. .. .. .. .. .. .. . . . . . . . . . . . . . . . AscoTlhie'osr em §4E.x erc. i.s e.s . . . .. .. . .. .. .. .. .. ... ...5 9.. .. .. .. .. .. .. .. .. .. .. . . . . .
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