HANCHEOLSEN AND HELGE HOLDEN Abstract We show that the Arzel57568Ascoli theorem and Kolmogorov com pactness theorem both are consequences of a simple lemma on compactness in metric spaces Their relation to Hellys theorem is discussed The paper cont ID: 69321
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2HANCHE-OLSENANDHOLDENHereisthekeylemmaformanycompactnessresults(inthislemmaanditsproof,everymetricisnamedd):Lemma1.LetXbeametricspace.Assumethat,forevery"�0,thereexistssome�0,ametricspaceW,andamapping:X!Wsothat[X]istotallybounded,andwheneverx;y2Xaresuchthatd�(x);(y),thend(x;y)".ThenXistotallybounded.Proof.Forany"&#x-278;0,pick,Wandasinthestatementofthelemma.Since[X]istotallybounded,thereexistsa nite-coverfV1;:::;Vngof[X].Thenitimmediatelyfollowsfromtheassumptionsthatf�1(V1);:::;�1(Vn)gisan"-coverofX.ThusXistotallybounded.Lemma1embodiesthemainargumentinthestandardproofoftheclassicalArzelàAscolitheorem,aswenowdemonstrate.Theorem2(ArzelàAscoli).Let beacompacttopologicalspace.ThenasubsetofC( )istotallyboundedinthesupremumnormif,andonlyif,(i)itispointwisebounded,and(ii)itisequicontinuous.Recallthede nitionofequicontinuity:Condition(ii)meansthatforeveryx2 andevery"&#x-278;0thereisaneighborhoodVofxsothatjf(y)�f(x)j"forally2Vandallfinthegivensetoffunctions.Proof.AssumeFC( )ispointwiseboundedandequicontinuous.Let"&#x-367;0.CombiningtheequicontinuityofFandcompactnessof ,wecan nda nitesetofpointsx1;:::;xn2 withneighborhoodsV1;:::;Vncoveringallof sothatjf(x)�f(xj)j"wheneverf2Fandx2Vj.De ne:F!Rnby(f)=�f(x1);:::;f(xn):BythepointwiseboundednessofF,theimage[F]isbounded,andhencetotallybounded,inRn.Furthermore,iff;g2Fwithk(f)�(g)k1",thensinceanyx2 belongstosomeVj,jf(x)�g(x)jjf(x)�f(xj)j+jf(xj)�g(xj)j+jg(xj)�g(x)j3";andsokf�gk13".ByLemma1,Fistotallybounded.Fortheconverse,assumethatFisatotallyboundedsubsetofC( ).Theexistenceofa nite"-coverforF,forany",clearlyimpliesthebound-ednessofF,thusestablishingtheuniformboundednessandhencealsopointwiseboundednessofF.Toproveequicontinuity,letx2 and"&#x]TJ/;༹ ; .96;& T; 10;&#x.516;&#x 0 T; [0;0begiven.Pickan"-coverfU1;:::;UngofF,andchosegj2Ujforj=1;:::;n.PickaneighborhoodVjofxsothatjgj(y)�gj(x)j"whenevery2Vj,forj=1;:::;n.LetV=V1\\Vm.Iff2Ujthenkf�gjk1",andsowheny2V,jf(y)�f(x)jjf(y)�gj(y)j+jgj(y)�gj(x)j+jgj(x)�f(x)j3";whichprovesequicontinuity.Remark3.Thistheoremwas rstprovedbyAscoli[3]forequi-LipschitzfunctionsandextendedbyArzelà[2]toageneralfamilyofequicontinuousfunctions.See[4,p.203]. THEKOLMOGOROVRIESZCOMPACTNESSTHEOREM3Wepresentthefollowingtheorem, rstprovedbyFréchet[12]forthecasep=2,asawarm-upexercise,astheproofisshortandnicelyexposessomekeyideasfortheproofofTheorem5.Theorem4.Asubsetoflp,where1p1,istotallyboundedif,andonlyif,(i)itispointwisebounded,and(ii)forevery"&#x]TJ/;གྷ ; .96;& T; 18;&#x.296;&#x 0 T; [0;0thereissomensothat,foreveryxinthegivensubset,Xk&#x]TJ/;གྷ ; .96;& T; 18;&#x.296;&#x 0 T; [0;njxkjp"p:Proof.AssumethatFlpsatis esthetwoconditions.Given"&#x-278;0,picknasinthesecondcondition,andde neamapping:F!Rnby(x)=(x1;:::;xn):BythepointwiseboundednessofF,theimage(F)istotallybounded.Ifx;y2Fwithj(x)�(y)jp=�Pnk=1jxk�ykjp1=p",thenkx�ykpnXk=1jxk�ykjp1=p+Xk&#x-278;njxk�ykjp1=p"+2"=3":ByLemma1,Fistotallybounded.Wewillleaveprovingtheconverseasanexercisetothereader.ThetechniquesfromtheproofofTheorem2areeasilyadapted.SeealsotheproofofTheorem5.3.TheKolmogorovRiesztheoremTheorem5(KolmogorovRiesz).Let1p1.AsubsetFofLp(Rn)istotallyboundedif,andonlyif,(i)Fisbounded,(ii)forevery"&#x]TJ/;གྷ ; .96;& T; 18;&#x.296;&#x 0 T; [0;0thereissomeRsothat,foreveryf2F,Zjxj&#x]TJ/;གྷ ; .96;& T; 18;&#x.296;&#x 0 T; [0;Rjf(x)jpdx"p;(iii)forevery"&#x-278;0thereissome&#x-278;0sothat,foreveryf2Fandy2Rnwithjyj,ZRnjf(x+y)�f(x)jpdx"p:Proof.AssumethatFLp(Rn)satis esthethreeconditions.First,given"&#x-278;0,pickRasinthesecondcondition,andasinthethirdcondition.LetQbeanopencubecenteredattheoriginsothatjyj1 2whenevery2Q.LetQ1;:::;QNbemutuallynon-overlappingtranslatesofQsothattheclosureofSiQicontainstheballwithradiusRcenteredattheorigin.LetPbetheprojectionmapofLp(Rn)ontothelinearspanofthecharacteristicfunctionsofthecubesQigivenbyPf(x)=8:1 jQijZQif(z)dz;x2Qi;i=1;:::;N;0otherwise:From(ii)andthede nitionofPfwe nd,forf2F,kf�Pfkpp"p+NXi=1ZQijf(x)�Pf(x)jpdx="p+NXi=1ZQi1 jQijZQi�f(x)�f(z)dzpdx: 6HANCHE-OLSENANDHOLDENProof.NotethatprecompactnessofFinLploc( )isequivalenttoprecompactnessofFK=ffK:f2FgforeverycompactK .Corollary9.AsubsetFWp;k(Rn)istotallyboundedif,andonlyif,thefol-lowingholds:(i)Fisbounded,i.e.,thereissomeMsothatZjD f(x)jpdxM;f2F;j jk:(ii)Forevery"&#x-278;0thereissomeRsothatZjxj&#x-278;RjD f(x)jpdx"p;f2F;j jk:(iii)Forevery"&#x-277;0thereissome&#x-277;0sothatZRnjD f(x+y)�D f(x)jpdx"p;f2F;j jk;jyj:Proof.NotethatFistotallyboundedinWp;k(Rn)ifandonlyifD [F]=fD f:f2FgistotallyboundedinLp(Rn)foreverymulti-index withj jk.4.AbitofhistoryIn1931,Kolmogorov[18]provedthe rstresultinthisdirection.ItcharacterizescompactnessinLp(Rn)for1p1,inthecasewhereallfunctionsaresupportedinacommonboundedset.Condition(iii)ofTheorem5isreplacedbytheuniformconvergenceinLpnormofsphericalmeansofeachfunctionintheclasstothefunctionitself.(Clearly,ourCondition(ii)isautomaticinthiscase.)Justayearlater,Tamarkin[26]expandedthisresulttothecaseofunboundedsupportsbyaddingCondition(ii)ofTheorem5.In1933,Tulajkov[29]expandedtheKolmogorovTamarkinresulttothecasep=1.Inthesameyear,andprobablyindependently,Riesz[23]provedtheresultfor1p1,essentiallyintheformofourTheorem5.Thuswefeelsomewhatjusti edinusingthenamesKolmogorovandRieszinreferringtothetheorem,thoughweareperhapsbeingabitunfairtoTamarkinandTulajkovindoingso.Thecompactnesstheoremhasalsoseengeneralizationsinotherdirections.Hanson[15]provedanecessaryandsu cientconditionforcompactnessofafamilyofmeasurablefunctionsonaboundedmeasurableset,withrespecttocon-vergenceinmeasure.(Herethemeasurablefunctionsformametricspaceinwhichthedistancebetweentwofunctionsisthein mumofall"&#x]TJ/;གྷ ; .96;& T; 21;&#x.726;&#x 0 T; [0;0sothatthetwofunctionsdierbyatmost"exceptonasetofmeasure".)Fréchet[13]replacedConditions(i)and(ii)ofTheorem5withasinglecondition(equisummability),andgeneralizedthetheoremtoarbitrarypositivep.Phillips[22,Thm3.7]provedanecessaryandsu cientconditionforcompactnessinLponageneralmeasurespace(1p1),andindeedinanyBanachspace,whichishoweversomewhatlesssuitedtoapplicationstoPDEs.Nevertheless,oursu ciencyproofforTheorem5isbasedonPhillips'criterion.(Itismorecommon,albeitmoreinvolved,tousemolli ersintheproof.)Weil[31](seealso[9,p.269])extendedtheresulttoLp(G)whereGisacompactgroup.Tsuji[28]consideredthecaseofLp(Rd)with0p1,andTakahashi[25]studiedthesameprobleminOrliczspaces.AcharacterizationofcompactsubsetsofLp([0;T];B)(BaBanachspace),whichisveryconvenientinthecontextoftime-dependentpartialdierentialequations,isgivenbySimon[24](seealso[19]).Areadableaccountofsomeofthehistoricaldevelopmentcanbe 10HANCHE-OLSENANDHOLDEN[18]A.N.Kolmogorov.ÜberKompaktheitderFunktionenmengenbeiderKonvergenzimMittel,Nachr.Ges.Wiss.Göttingen9(1931),6063.Englishtranslation:Onthecompactnessofsetsoffunctionsinthecaseofconvergenceinthemean,inV.M.Tikhomirov(ed.),SelectedWorksofA.N.Kolmogorov,Vol.I,Kluwer,Dordrecht,1991,pp.147150.[19]E.Maitre.OnanonlinearcompactnesslemmainLp(0;T;B).Int.J.Math.Math.Sci.no.27(2003)17251730.[20]M.Nicolescu.OnthecriterionofcompactnessofA.Kolmogorov.(InRomanian)Acad.Repub.Pop.Române.Bul.“ti.Ser.Math.Fiz.Chim.2(1950)407415.[21]R.L.Pego.CompactnessinL2andtheFourierTransform.Proc.Amer.Math.Soc.95(1985)252254.[22]R.S.Phillips.Onlineartransforms.Trans.Amer.Math.Soc.48(1940)516541.[23]M.Riesz.Surlesensemblescompactsdefonctionssommables.(InFrench)ActaSzegedSect.Math.6(1933),136142.AlsoinL.Gårding,L.Hörmander(eds.),MarcelRieszCollectedPapers,Springer,Berlin(1988).[24]J.Simon.CompactsetsinthespaceLp(0;T;B).Ann.Mat.PuraAppl.146(1987),6596.[25]T.Takahashi.Onthecompactnessofthefunction-setbytheconvergenceinmeanofgeneraltype.StudioMath.5(1934)141150.[26]J.D.Tamarkin.OnthecompactnessofthespaceLp.Bull.Amer.Math.Soc.32(1932),7984.[27]Yu.V.Tret'yachenkoandV.V.Chistyakov.Selectionprincipleforpointwiseboundedse-quencesoffunctions.Math.Notes84(2008)396406.[28]M.Tsuji.OnthecompactnessofspaceLp(p�0)anditsapplicationtointegraloperators.KodaiMath.J.(1951)3336.[29]A.Tulajkov.ZurKompaktheitimRaumLpfürp=1.(InGerman)Nachr.Ges.Wiss.Göttingen,Math.Phys.Kl.I1933,nr.39,167170.[30]P.Veress.ÜberFunctionenmengen.(InGerman)Actascient.math.3(1927)177192.[31]A.Weil.L'intégrationdanslesgroupestopologiquesetsesapplications.HermannetCie.,Paris,1940.[32]K.Yosida.FunctionalAnalysis.Springer,Berlin,1980.(Hanche-Olsen)DepartmentofMathematicalSciencesNorwegianUniversityofScienceandTechnologyNO7491Trondheim,NorwayE-mailaddress:hanche@math.ntnu.noURL:http://www.math.ntnu.no/hanche/(Holden)DepartmentofMathematicalSciencesNorwegianUniversityofScienceandTechnologyNO7491Trondheim,Norway,andCentreofMathematicsforApplicationsUniversityofOsloP.O.Box1053,BlindernNO0316Oslo,NorwayE-mailaddress:holden@math.ntnu.noURL:http://www.math.ntnu.no/holden/