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Real and Convex Analysis_Springer_2013.pdf

Real and Convex Analysis_Spring…

上传者: terminaqz 2014-03-11 评分 4.5 0 87 12 397 暂无简介 简介 举报

简介:本文档为《Real and Convex Analysis_Springer_2013pdf》,可适用于高等教育领域,主题内容包含ErhanCınlar•RobertJVanderbeiRealandConvexAnalysisErhanCınlarDepartmentofOp符等。

ErhanCınlar•RobertJVanderbeiRealandConvexAnalysisErhanCınlarDepartmentofOperationsResearchandFinancialEngineeringPrincetonUniversityPrinceton,NewJerseyUSARobertJVanderbeiDepartmentofOperationsResearchandFinancialEngineeringPrincetonUniversityPrinceton,NewJerseyUSAISSNISBNISBN(ebook)DOISpringerNewYorkHeidelbergDordrechtLondonLibraryofCongressControlNumber:MathematicsSubjectClassification:MSC:,,,,,,cSpringerScienceBusinessMediaNewYorkThisworkissubjecttocopyrightAllrightsarereservedbythePublisher,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedExemptedfromthislegalreservationarebriefexcerptsinconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaseroftheworkDuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringerPermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenterViolationsareliabletoprosecutionundertherespectiveCopyrightLawTheuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etcinthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluseWhiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforanyerrorsoromissionsthatmaybemadeThepublishermakesnowarranty,expressorimplied,withrespecttothematerialcontainedhereinPrintedonacidfreepaperSpringerispartofSpringerScienceBusinessMedia(wwwspringercom)PrefaceThisbookisintendedtoserveasafirstcourseinanalysisforscientistsandengineersItcanbeusedeitherattheadvancedundergraduateleveloraspartofthecurriculuminagraduateprogramWehavetaughtfrompreliminarydraftsofthebookforseveralyearsThebookisbuiltaroundmetricspacesInthefirstthreechapters,welaythefoundationalmaterialWecovertheallimportant“fourCs”:convergence,completeness,compactness,andcontinuityWehaveorganizedthematerialtobeassimpleandaslogicalaspossibleInsubsequentchapters,weusethebasictoolsofanalysistogiveabriefintroductiontocloselyrelatedtopicssuchasdifferentialandintegralequations,convexanalysis,andmeasuretheoryThebookisshortandyetcoversinsomedepththemostimportantsubjectsWegavecarefulconsiderationtowhattoincludeandwhattoleaveoutInallsuchconsiderations,weaskedourselveswhetherthematerialwouldbeofdirectandimmediateusetoscientistsandengineersOurphilosophyis“ifindoubt,dowithout”WhatmakesthisbookdifferentWepulltogethersomeofthefoundationalmaterialonemightfind,forexample,intheclassicbookbyRudinRudwithmaterialonconvexityandoptimizationatalevelcommensurate,say,withthebookbyBorweinandLewisBLandwithacompletelymoderntreatmentofthebasicsofmeasuretheoryTheimportanceofmeasuretheoryhasincreasedovertheyearsasstochasticmodelinghasbecomemorecentraltoallaspectsofanalysisSimilarly,optimizationplaysaneverincreasingroleasonetriestodesignandanalyzethebestpossible“widget”WehopethatthereaderwillenjoythebookandlearnsomeimportantmathematicsWewouldliketothankthemanystudentswhomwehavehadthepleasureofteachingovertheyearsWegiveaspecialthankstoJohnD’AngelohecarefullyreadadraftofthemanuscriptandmadenumeroushelpfulsuggestionsECınlarandRJVanderbeivContentsPrefacevNotationandUsageixChapterSetsandFunctionsASetsBFunctionsandSequencesCCountabilityDOntheRealLineESeriesChapterMetricSpacesAEuclideanSpacesBMetricsCOpenandClosedSetsDConvergenceECompletenessFCompactnessChapterFunctionsonMetricSpacesAContinuousMappingsBCompactnessandUniformContinuityCSequencesofFunctionsDSpacesofContinuousFunctionsChapterDifferentialandIntegralEquationsAContractionMappingsBSystemsofLinearEquationsCIntegralEquationsDDifferentialEquationsChapterConvexityAConvexSetsandConvexFunctionsBProjectionsviiviiiContentsCSupportingHyperplaneTheoremDLegendreTransformEInfimalConvolutionChapterConvexOptimizationAPrimalandDualProblemsBLinearProgrammingandPolyhedraCLagrangiansDSaddlePointsChapterMeasureandIntegrationAAlgebrasBMeasurableSpacesandFunctionsCMeasuresDIntegrationETransformsandIndefiniteIntegralsFKernelsandProductSpacesFurtherReadingBibliographyIndexNotationandUsageWeusetheterms“positive”and“negative”intheirwidesense:positivemeans,negativemeansSimilarly,“increasing”meansxyimpliesf(x)f(y)Ifstrictinequalitieshold,wesay“strictlypositive,”“strictlynegative,”“strictlyincreasing,”etcHereisalistoffrequentlyusednotations:TheemptysetN={,,,}:ThesetofnaturalnumbersN={,,,}:ThesetofstrictlypositiveintegersZ={,,,,,}:ThesetofintegersQ={x:x=mnforsomeminZandsomeninN}:ThesetofrationalsR=(,)={x:<x<}:ThesetofrealsR=,)={xR:x}:ThesetofpositiverealsR=(,:ThesetofrealsandplusinfinityR=,:Thesetofextendedrealsa,b={xR:axb}:Theclosedintervalwithendpointsaandb(a,b)={xR:a<x<b}:Theopenintervalwithendpointsaandblog(x):Thenaturallogarithmofxxy=nxiyi:Theinnerproductofxandy‖x‖=xx:TheEuclideannormofxC:Thesetofcontinuousfunctionsd(x,y):ThedistancefromxtoyB(x,r)={y:d(x,y)<r}:TheopenballcenteredonxofradiusrA:TheinteriorofthesetAA:TheclosureofthesetAA:TheboundaryofthesetAfˆ:TheLegendretransformofthefunctionffg:Theinfimalconvolutionofthefunctionsfandgxy:Theminimumoftherealnumbersxandyxy:ThemaximumoftherealnumbersxandyA:TheindicatorfunctionofthesetAB(E):TheBorelσalgebraonthemetricspaceEixCHAPTERSetsandFunctionsThisintroductorychapterisdevotedtogeneralnotionsregardingsets,functions,sequences,andseriesWeaimtointroduceandreviewthebasicnotation,terminology,conventions,andelementaryfactsASetsAsetisacollectionofsomeobjectsGivenaset,theobjectsthatformitarecalleditselementsGivenasetA,wewritexAtomeanthatxisanelementofATosaythatxA,wealsosayxisinA,orxisamemberofA,orxbelongstoA,orAincludesxOnecanspecifyasetbylistingitselementsinsidecurlybraces,butdoingsoisnotfeasibleinmostcasesMoreoftenwespecifyasetbypreciselydescribingitselementsForexample,A={a,b,c}isthesetwhoseelementsarea,b,andc,andB={x:x>}isthesetofallnumbersexceedingThefollowingaresomespecialsets::TheemptysetIthasnoelementsN={,,,}:ThesetofnaturalnumbersN={,,,}:ThesetofstrictlypositiveintegersZ={,,,,,}:ThesetofintegersQ={x:x=mnforsomeminZandsomeninN}:ThesetofrationalsR=(,)={x:<x<}:ThesetofrealsR=,)={xR:x}:Thesetofpositiverealsa,b={xR:axb}:Theclosedintervalwithendpointsaandb(a,b)={xR:a<x<b},definedfornumbersaandbwitha<b:TheopenintervalwithendpointsaandbWeassumethatthereaderisfamiliarwiththesesetsForexample,wetakeitforgrantedthatrealnumbersarelimitsofrationalsSubsetsAsetAissaidtobeasubsetofasetBifeveryelementofAisanelementofBWewriteABorBAtoindicatethisandsayAiscontainedinB,orBcontainsA,tothesameeffectThesetsAandBarethesameifandonlyifABandAB,ECınlarandRJVanderbei,RealandConvexAnalysis,UndergraduateTextsinMathematics,DOI,SpringerScienceBusinessMediaNewYorkSetsandFunctionsChapandthenwewriteA=BForthecontrarysituations,wewriteA=BwhenAandBarenotthesameThesetAiscalledapropersubsetofBifAisasubsetofB,andAandBarenotthesameTheemptysetisasubsetofeverysetLetAbeasetTheclaimisthatA,thatis,thateveryelementofisalsoanelementofA,orequivalently,thereisnoelementofthatdoesnotbelongtoAButthelastphraseistruesimplybecausehasnoelementsSetOperationsLetAandBbesetsTheirunion,denotedbyAB,isthesetconsistingofallelementsthatbelongtoeitherAorB(orboth)Theirintersection,denotedbyAB,isthesetofallelementsthatbelongtobothAandBThecomplementofAinB,denotedbyBA,isthesetofallelementsofBthatarenotinASometimes,whenBisunderstoodfromcontext,BAisalsocalledthecomplementofAandisdenotedbyAcRegardingtheseoperations,thefollowingstatementshold:Commutativelaws:AB=BA,AB=BAAssociativelaws:(AB)C=A(BC),(AB)C=A(BC)Distributivelaws:A(BC)=(AB)(AC),A(BC)=(AB)(AC)TheassociativelawsshowthatABCandABChaveunambiguousmeaningsDefinitionsofunionsandintersectionscanbeextendedtoarbitrarycollectionsofsetsLetIbeasetForeachiinI,letAibeasetTheunionofthesetsAi,iI,isthesetAsuchthatxAifandonlyifxAiforsomeiinITheintersectionofthemisthesetAsuchthatxAifandonlyifxAiforeveryiinIThefollowingnotationsareusedtodenotetheunionandintersectionrespectively:iIAi,iIAiWhenI={,,,},itiscustomarytowritei=Ai,i=AiSecASetsAllofthesenotationsfollowtheconventionsforsumsofnumbersForinstance,ni=Ai=AAn,i=Ai=AAADisjointSetsTwosetsaresaidtobedisjointiftheirintersectionisemptythatis,iftheyhavenoelementsincommonAcollection{Ai:iI}ofsetsissaidtobedisjointedifAiandAjaredisjointforeveryiandjinIwithi=jProductsofSetsLetAandBbesetsTheirproduct,denotedbyAB,isthesetofallpairs(x,y)withxinAandyinBItisalsocalledtherectanglewithsidesAandBIfA,,Anaresets,thentheirproductAAnisthesetofallntuples(x,,xn)wherexA,,xnAnThisproductiscalled,variously,arectangle,orabox,oranndimensionalboxIfA==An=A,thenAAnisdenotedbyAnThus,Ristheplane,Risthethreedimensionalspace,Risthepositivequadrantoftheplane,etcExercisesLetEbeasetShowthefollowingforsubsetsA,B,C,andAiofEHere,allcomplementsarewithrespecttoEforinstance,Ac=EA(a)(Ac)c=A(b)BA=BAc(c)(BA)C=(BC)(AC)(d)(AB)c=AcBc(e)(AB)c=AcBc(f)(iIAi)c=iIAci(g)(iIAi)c=iIAciLetaandbberealnumberswitha<bFindn=ïan,bnò,n=ïan,bnòDescribethefollowingsetsinwordsandpictures:(a)A={xR:xx<}SetsandFunctionsChap(b)B={xR:xx}(c)C=BA(d)D=CB(e)S=CCLetAnbethesetofpoints(x,y)inRlyingonthecurvey=xn,<x<WhatisnAnBFunctionsandSequencesLetEandFbesetsWitheachelementxofE,lettherebeassociatedauniqueelementf(x)ofFThenfiscalledafunctionfromEintoF,andfissaidtomapEintoFWewritef:EFtoindicatethisLetfbeafunctionfromEintoFForxinE,thepointf(x)inFiscalledtheimageofxorthevalueoffatxSimilarly,forAE,theset{yF:y=f(x)forsomexA}iscalledtheimageofAInparticular,theimageofEiscalledtherangeoffMovingintheoppositedirection,forBF,fB={xE:f(x)B}iscalledtheinverseimageofBunderfObviously,theinverseimageofFisETermslikemapping,operator,transformationaresynonymsfortheterm“function”withvaryingshadesofmeaningdependingonthecontextandonthesetsEandFWeshallbecomefamiliarwiththemintimeSometimes,wewritexf(x)toindicatethemappingfforinstance,themappingxxfromRintoRisthefunctionf:RRdefinedbyf(x)=xInjections,Surjections,BijectionsLetfbeafunctionfromEintoFItiscalledaninjection,orissaidtobeinjective,orissaidtobeonetoone,ifdistinctpointshavedistinctimages,thatis,ifx=yimpliesf(x)=f(y)Itiscalledasurjection,orissaidtobesurjective,ifitsrangeisF,inwhichcasefissaidtobefromEontoFItiscalledabijection,orissaidtobebijective,ifitisbothinjectiveandsurjectiveThesetermsarerelativetoEandFForexample,xexisaninjectionfromRintoR,butisabijectionfromRonto(,)ThefunctionxsinxfromRintoRisneitherinjectivenorsurjective,butitisasurjectionfromRonto,SequencesAsequenceisafunctionfromNintosomesetIffisasequence,itiscustomarytodenotef(n)bysomethinglikexnandwrite(xn)or(x,x,)forthesequenceSecBFunctionsandSequences(insteadoff)Then,thexnarecalledthetermsofthesequenceForinstance,(,,,,,)isasequencewhosefirst,second,etctermsarex=,x=,etcSometimesitisconvenienttodefineasequenceoverN,andthenwrite(xn)nNor(x,x,)foritIfAisasetandeverytermofthesequence(xn)belongstoA,then(xn)issaidtobeasequenceinAorasequenceofelementsofA,andwewrite(xn)AtoindicatethiswithaslightabuseofnotationAsequence(xn)issaidtobeasubsequenceof(yn)ifthereexistintegersk<k<k<suchthatxn=yknforeachnForinstance,thesequence(,,,,)isasubsequenceof(,,,,,)ExercisesInverseimagesLetfbeamappingfromEintoFShowthat(a)f=,(b)fF=E,(c)f(BC)=(fB)(fC),(d)f(iIBi)=iIfBi,(e)f(iIBi)=iIfBi,forallsubsetsB,C,BiofFExponentialandlogarithmShowthatxexisabijectionfromRonto(,Showthatxlogxisabijectionfrom(,)ontoR(Incidentally,logxisthelogarithmofxtothebasee,whichisnowadayscalledthenaturallogarithmWecallitthelogarithmLetotherscalltheirlogarithms“unnatural”and,whiletheyareatit,theycanalsoinventunnaturalexponentialslikexax)BijectionsbetweenNandZLetfbedefinedbythearrowsbelow(forinstance,f()=):ThisdefinesabijectionfromNontoZUsingthis,constructabijectionfromZontoNSetsandFunctionsChapBijectionfromZZontoNLetf:NNNbedefinedbythetablebelowwheref(i,j)istheentryintheithrowandthejthcolumnUsethisandtheprecedingexercisetoconstructabijectionfromZZontoNjiFunctionalinversesLetfbeabijectionfromEontoFThen,foreachyinFthereisauniquexinEsuchthatf(x)=yInotherwords,inthenotationof,f({y})={x}foreachyinFandsomeuniquexinEInthiscase,wedropsomebracketsandwritef(y)=xTheresultingfunctionisabijectionfromFontoEitiscalledthefunctionalinverseoffThisparticularusageshouldnotbeconfusedwiththegeneralnotationoff(NotethatdefinesafunctionffromFintoE,whereFisthecollectionofallsubsetsofFandEisthecollectionofallsubsetsofE)CCountabilityTwosetsAandBaresaidtohavethesamecardinalityifthereexistsabijectionfromAontoB,andthenwewriteABObviously,havingthesamecardinalityisanequivalencerelation:itis(a)reflexive,AA(b)symmetric,ABBA(c)transitive,ABandBCACAsetissaidtobefiniteifitisemptyorhasthesamecardinalityas{,,,n}forsomeninNintheformercaseithaselements,inthelatterexactlynItissaidtobecountableifitisfiniteorhasthesamecardinalityasNinthelattercaseitissaidtohaveacountableinfinityofelementsInparticular,NiscountableSoareZandNNinviewofExercisesandNotethataninfinitesetcanhavethesamecardinalityasoneofitspropersubsetsForinstance,ZN,R(,,RR(,)seeExerciseforthelatterIncidentally,R,R,etcareuncountable,asweshallshowshortlySecCCountabilityAsetiscountableifandonlyifitcanbeinjectedintoN,orequivalently,ifandonlyifthereisasurjectionfromNontoitThus,asetAiscountableifandonlyifthereisasequence(xn)whoserangeisAThefollowinglemmafollowseasilyfromtheseremarksLEMMAIfAcanbeinjectedintoB,andifBiscountable,thenAiscountableIfAiscountableandthereisasurjectionfromAontoB,thenBiscountableTHEOREMTheproductoftwocountablesetsiscountablePROOFLetAandBbecountableIfoneofthemisempty,thenABisemptyandthereisnothingtoproveSupposethatneitherisemptyThen,thereexistinjectionsf:ANandg:BNForeachpair(x,y)inAB,leth(x,y)=(f(x),g(y))thenhisaninjectionfromABintoNNSinceNNiscountable(seeExercise),thisimpliesviatheprecedinglemmathatABiscountableCOROLLARYThesetofallrationalnumbersiscountablePROOFRecallthateachrationalisaratiomnwithminZandninNThus,f(m,n)=mndefinesasurjectionfromZNontothesetQofallrationalsSinceZandNarecountable,soisZNbytheprecedingtheoremHence,QiscountablebyLemmaTHEOREMTheunionofacountablecollectionofcountablesetsiscountablePROOFLetIbeacountableset,andletAibeacountablesetforeachiinITheclaimisthatA=iIAiiscountableNow,thereisasurjectionfi:NAiforeachi,andthereisasurjection

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