Lectures on Functional Analysis and the Lebesgue Integral PDF
Preview Lectures on Functional Analysis and the Lebesgue Integral
Universitext Vilmos Komornik Lectures on Functional Analysis and the Lebesgue Integral Universitext Universitext SeriesEditors SheldonAxler SanFranciscoStateUniversity VincenzoCapasso UniversitàdegliStudidiMilano CarlesCasacuberta UniversitatdeBarcelona AngusMacIntyre QueenMary,UniversityofLondon KennethRibet UniversityofCalifornia,Berkeley ClaudeSabbah CNRS,ÉcolePolytechnique,Paris EndreSüli UniversityofOxford WojborA.Woyczyn´ski CaseWesternReserveUniversityCleveland,OH Universitext is a series of textbooksthat presents material from a wide variety of mathematicaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass- testedbytheirauthor,mayhaveaninformal,personalevenexperimentalapproach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teachingcurricula,toverypolishedtexts. Thus as research topics trickle down into graduate-level teaching, first textbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. Moreinformationaboutthisseriesathttp://www.springer.com/series/223 Vilmos Komornik Lectures on Functional Analysis and the Lebesgue Integral 123 VilmosKomornik UniversityofStrasbourg Strasbourg,France TranslationfromtheFrenchlanguageedition: Précisd’analyseréelle-Analyse fonctionnelle,intégraledeLebesgue,espaces fonctionnels,vol-2byVilmosKomornik Copyright©2002EditionMarketingS.A. www.editions-ellipses.fr/ AllRightsReserved ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-1-4471-6810-2 ISBN978-1-4471-6811-9 (eBook) DOI10.1007/978-1-4471-6811-9 LibraryofCongressControlNumber:2016941752 Mathematics Subject Classification: 46-01, 46E10, 46E15, 46E20, 28-01, 28A05, 28A20, 28A25, 28A35,41A10,41A36 ©Springer-VerlagLondon2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringer-VerlagLondonLtd. Preface ThisbookisbasedonlecturesgivenbytheauthorattheUniversityofStrasbourg. Functionalanalysisispresentedfirst,inanontraditionalway:wetrytogeneral- izesomeelementarytheoremsofplanegeometrytospacesofarbitrarydimension. This approach leads us to the basic notions and theorems in a natural way. The resultsareillustratedinthesmall`pspaces. The Lebesgue integral is treated next by following F. Riesz. Starting with two innocent-looking lemmas on step functions, the whole theory is developed in a surprisingly short and clear manner. His constructive definition of measurable functions quickly leads to optimal versions of the classical theorems of Fubini– TonelliandRadon–Nikodým. These two parts are essentially independent of each other, and only basic topological results are used. In the last part, they are combined to study various functionspacesofcontinuousandintegrablefunctions. Weindicatetheoriginalsourcesofmostnotionsandresults.Someothernovelties arementionedonpage375.Thematerialmarkedbythesymbol(cid:2)maybeskipped duringthefirstreading. Eachchapterendswithalistofexercises.However,themostimportantexercises areincorporatedinthetextasexamplesandremarks,andthereaderisexpectedto fillinthemissingdetails. Welistonp.xisomeinterestingpapersofthegeneralmathematicalculture. We have put a great deal of effort into selecting the material, formulating aestheticandgeneralstatements,seekingshortandelegantproofs,andillustrating theresultswithsimplebutpertinentexamples.Ourworkwasstronglyinfluencedby thebeautifullecturesofÁ.CsászárandL.CzáchattheEötvösLorándUniversity, Budapest,inthe1970s,andmoregenerallybytheHungarianmathematicaltradition createdbyLeopoldFejér,FrédéricRiesz,PaulTurán,PaulErdo˝s,andothers. v vi Preface We also thank C. Baud, B. Beeton, Á. Besenyei, T. Delzant, C. Disdier, O. Gebuhrer, V. Kharlamov, P. Loreti, C.-M. Marle, P. Martinez, P.P. Pálfy, P. Pilibossian, J. Saint Jean Paulin, Z. Sebestyén, A. Simonovits, Mrs B. Szénássy, J.Vancostenoble,andtheeditorsofSpringerfortheirprecioushelp. Thisbookisdedicatedtothememoryofmyfather. Strasbourg,France VilmosKomornik May23,2016 Contents PartI FunctionalAnalysis 1 HilbertSpaces............................................................... 3 1.1 DefinitionsandExamples............................................ 3 1.2 Orthogonality......................................................... 11 1.3 SeparationofConvexSets:TheoremsofRiesz–Fréchet andKuhn–Tucker .................................................... 16 1.4 OrthonormalBases................................................... 24 1.5 WeakConvergence:TheoremofChoice............................ 29 1.6 ContinuousandCompactOperators................................. 35 1.7 Hilbert’sSpectralTheorem.......................................... 39 1.8 *TheComplexCase................................................. 45 1.9 Exercises.............................................................. 47 2 BanachSpaces .............................................................. 55 2.1 SeparationofConvexSets........................................... 57 2.2 TheoremsofHelly–Hahn–BanachandTaylor–Foguel............. 65 2.3 The`pSpacesandTheirDuals...................................... 69 2.4 BanachSpaces........................................................ 76 2.5 WeakConvergence:Helly–Banach–SteinhausTheorem........... 79 2.6 ReflexiveSpaces:TheoremofChoice............................... 87 2.7 ReflexiveSpaces:GeometricalApplications........................ 91 2.8 *OpenMappingsandClosedGraphs............................... 96 2.9 *ContinuousandCompactOperators............................... 99 2.10 *Fredholm–RieszTheory ........................................... 103 2.11 *TheComplexCase................................................. 112 2.12 Exercises.............................................................. 113 3 LocallyConvexSpaces..................................................... 119 3.1 FamiliesofSeminorms............................................... 120 3.2 SeparationandExtensionTheorems ................................ 123 3.3 Krein–MilmanTheorem............................................. 126 vii viii Contents 3.4 *WeakTopology.Farkas–MinkowskiLemma ..................... 130 3.5 *WeakStarTopology:TheoremsofBanach–Alaoglu andGoldstein......................................................... 135 3.6 *ReflexiveSpaces:TheoremsofKakutani andEberlein–Šmulian................................................ 140 3.7 *TopologicalVectorSpaces......................................... 144 3.8 Exercises.............................................................. 146 PartII TheLebesgueIntegral 4 *MonotoneFunctions ..................................................... 151 4.1 Continuity:CountableSets .......................................... 151 4.2 Differentiability:NullSets........................................... 154 4.3 JumpFunctions....................................................... 157 4.4 ProofofLebesgue’sTheorem ....................................... 161 4.5 FunctionsofBoundedVariation..................................... 164 4.6 Exercises.............................................................. 165 5 TheLebesgueIntegralinR................................................ 169 5.1 StepFunctions........................................................ 170 5.2 IntegrableFunctions.................................................. 174 5.3 TheBeppoLeviTheorem............................................ 177 5.4 TheoremsofLebesgue,FatouandRiesz–Fischer .................. 181 5.5 *MeasurableFunctionsandSets.................................... 187 5.6 Exercises.............................................................. 194 6 *GeneralizedNewton–LeibnizFormula................................. 197 6.1 AbsoluteContinuity.................................................. 198 6.2 PrimitiveFunction.................................................... 203 6.3 IntegrationbyPartsandChangeofVariable........................ 207 6.4 Exercises.............................................................. 209 7 IntegralsonMeasureSpaces.............................................. 211 7.1 Measures.............................................................. 211 7.2 IntegralsAssociatedwithaFiniteMeasure......................... 217 7.3 ProductSpaces:TheoremsofFubiniandTonelli................... 224 7.4 SignedMeasures:HahnandJordanDecompositions .............. 229 7.5 LebesgueDecomposition............................................ 235 7.6 TheRadon–NikodýmTheorem...................................... 239 7.7 *LocalMeasurability................................................ 247 7.8 Exercises.............................................................. 251 PartIII FunctionSpaces 8 SpacesofContinuousFunctions.......................................... 257 8.1 WeierstrassApproximationTheorems .............................. 260 8.2 *TheStone–WeierstrassTheorem.................................. 265 Contents ix 8.3 CompactSets.TheArzelà–AscoliTheorem........................ 268 8.4 DivergenceofFourierSeries ........................................ 270 8.5 SummabilityofFourierSeries.Fejér’sTheorem................... 275 8.6 *Korovkin’sTheorems.BernsteinPolynomials.................... 279 8.7 *TheoremsofHaršiladze–Lozinski,NikolaevandFaber ......... 284 8.8 *DualSpace.RieszRepresentationTheorem...................... 289 8.9 WeakConvergence................................................... 299 8.10 Exercises.............................................................. 300 9 SpacesofIntegrableFunctions............................................ 305 9.1 Lp Spaces,1(cid:3)p(cid:3)1............................................... 305 9.2 *CompactSets....................................................... 316 9.3 *Convolution......................................................... 320 9.4 UniformlyConvexSpaces........................................... 323 9.5 Reflexivity............................................................ 329 9.6 DualsofLpSpaces................................................... 331 9.7 WeakandWeakStarConvergence.................................. 336 9.8 Exercises.............................................................. 339 10 AlmostEverywhereConvergence......................................... 341 10.1 Lp Spaces,1(cid:3)p(cid:3)1............................................... 341 10.2 Lp Spaces,0<p(cid:3)1................................................. 344 10.3 L0 Spaces ............................................................. 351 10.4 ConvergenceinMeasure............................................. 355 HintsandSolutionstoSomeExercises ........................................ 363 TeachingRemarks ............................................................... 375 Bibliography...................................................................... 377 SubjectIndex..................................................................... 395 NameIndex....................................................................... 401
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