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Texts and Readings in Mathematics 38 Terence Tao Analysis II Fourth Edition Texts and Readings in Mathematics AdvisoryEditor C.S.Seshadri,ChennaiMathematicalInstitute,Chennai,India ManagingEditor RajendraBhatia,AshokaUniversity,Sonepat,India EditorialBoard ManindraAgrawal,IndianInstituteofTechnology,Kanpur,India V.Balaji,ChennaiMathematicalInstitute,Chennai,India R.B.Bapat,IndianStatisticalInstitute,NewDelhi,India V.S.Borkar,IndianInstituteofTechnology,Mumbai,India ApoorvaKhare,IndianInstituteofScience,Bangalore,India T.R.Ramadas,ChennaiMathematicalInstitute,Chennai,India V.Srinivas,TataInstituteofFundamentalResearch,Mumbai,India TechnicalEditor P.Vanchinathan,VelloreInstituteofTechnology,Chennai,India TheTextsandReadingsinMathematicsseriespublisheshigh-qualitytextbooks, research-level monographs, lecture notes and contributed volumes. Undergraduate and graduate students of mathematics, research scholars and teachers would find thisbookseriesuseful.Thevolumesarecarefullywrittenasteachingaidsandhigh- lightcharacteristicfeaturesofthetheory.Booksinthisseriesareco-publishedwith HindustanBookAgency,NewDelhi,India. Terence Tao Analysis II Fourth Edition TerenceTao DepartmentofMathematics UniversityofCaliforniaLosAngeles LosAngeles,CA,USA ISSN 2366-8717 ISSN 2366-8725 (electronic) TextsandReadingsinMathematics ISBN ISBN 978-981-19-7284-3 (eBook) https://doi.org/10.1007/978-981-19-7284-3 JointlypublishedwithHindustanBookAgency Thisworkisaco-publicationwithHindustanBookAgency,NewDelhi,licensedforsaleinallcountries inelectronicformonly.SoldanddistributedinprintacrosstheworldbyHindustanBookAgency,P-19 GreenParkExtension,NewDelhi110016,India. ISBN:978-81-951961-2-8©HindustanBookAgency2022 Springerhasonlyelectronicright 3rdedition:©SpringerScience+BusinessMediaSingapore2016andHindustanBookAgency2015 4thedition:©HindustanBookAgency2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublishers,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishersnortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublishersremainneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Tomyparents,foreverything Preface to the First Edition ThistextoriginatedfromthelecturenotesIgaveteachingthehonoursundergraduate- level real analysis sequence at the University of California, Los Angeles, in 2003. Amongtheundergraduateshere,realanalysiswasviewedasbeingoneofthemost difficultcoursestolearn,notonlybecauseoftheabstractconceptsbeingintroduced forthefirsttime(e.g.,topology,limits,measurability,etc.),butalsobecauseofthe level of rigour and proof demanded of the course. Because of this perception of difficulty,onewasoftenfacedwiththedifficultchoiceofeitherreducingthelevel ofrigourinthecourseinordertomakeiteasier,ortomaintainstrictstandardsand facetheprospectofmanyundergraduates,evenmanyofthebrightandenthusiastic ones,strugglingwiththecoursematerial. Faced with this dilemma, I tried a somewhat unusual approach to the subject. Typically, an introductory sequence in real analysis assumes that the students are alreadyfamiliarwiththerealnumbers,withmathematicalinduction,withelementary calculus,andwiththebasicsofsettheory,andthenquicklylaunchesintotheheart of the subject, for instance the concept of a limit. Normally, students entering this sequencedoindeedhaveafairbitofexposuretotheseprerequisitetopics,though inmostcasesthematerialisnotcovered inathoroughmanner.Forinstance,very fewstudentswereabletoactuallydefinearealnumber,orevenaninteger,properly, even though they could visualize these numbers intuitively and manipulate them algebraically. This seemed to me to be a missed opportunity. Real analysis is one ofthefirstsubjects(togetherwithlinearalgebraandabstractalgebra)thatastudent encounters,inwhichonetrulyhastograpplewiththesubtletiesofatrulyrigorous mathematicalproof.Assuch,thecourseofferedanexcellentchancetogobackto thefoundationsofmathematics,andinparticulartheopportunitytodoaproperand thoroughconstructionoftherealnumbers. Thus the course was structured as follows. In the first week, I described some well-known “paradoxes” in analysis, in which standard laws of the subject (e.g., interchangeoflimitsandsums,orsumsandintegrals)wereappliedinanon-rigorous waytogivenonsensicalresultssuchas0=1.Thismotivatedtheneedtogobackto theverybeginningofthesubject,eventotheverydefinitionofthenaturalnumbers, andcheckallthefoundationsfromscratch.Forinstance,oneofthefirsthomework vii viii PrefacetotheFirstEdition assignmentswastocheck(usingonlythePeanoaxioms)thatadditionwasassociative for natural numbers (i.e., that (a+b)+c = a +(b+c) for all natural numbers a, b, c: see Exercise 2.2.1). Thus even in the first week, the students had to write rigorous proofs using mathematical induction. After we had derived all the basic propertiesofthenaturalnumbers,wethenmovedontotheintegers(initiallydefined asformaldifferencesofnaturalnumbers);oncethestudentshadverifiedallthebasic propertiesoftheintegers,wemovedontotherationals(initiallydefinedasformal quotientsofintegers);andthenfromtherewemovedon(viaformallimitsofCauchy sequences)tothereals.Aroundthesametime,wecoveredthebasicsofsettheory, forinstancedemonstratingtheuncountabilityofthereals.Onlythen(afteraboutten lectures)didwebeginwhatonenormallyconsiderstheheartofundergraduatereal analysis—limits,continuity,differentiability,andsoforth. The response to this format was quite interesting. In the first few weeks, the students found the material very easy on a conceptual level, as we were dealing onlywiththebasicpropertiesofthestandardnumbersystems.Butonanintellectual level it was very challenging, as one was analyzing these number systems from a foundationalviewpoint,inordertorigorouslyderivethemoreadvancedfactsabout thesenumbersystemsfromthemoreprimitiveones.Onestudenttoldmehowdiffi- cult it was to explain to his friends in the non-honours real analysis sequence (a) why he was still learning how to show why all rational numbers are either posi- tive,negative,orzero(Exercise4.2.4),whilethenon-honourssequencewasalready distinguishingabsolutelyconvergentandconvergentseries,and(b)why,despitethis, hethoughthishomeworkwassignificantlyharderthanthatofhisfriends.Another studentcommentedtome,quitewryly,thatwhileshecouldobviouslyseewhyone couldalwaysdivideanaturalnumbern intoapositiveintegerq togiveaquotient a and a remainder r less than q (Exercise 2.3.5), she still had, to her frustration, much difficulty in writing down a proof of this fact. (I told her that later in the course she would have to prove statements for which it would not be as obvious to see that the statements were true; she did not seem to be particularly consoled bythis.)Nevertheless,thesestudentsgreatlyenjoyedthehomework,aswhenthey didperservereandobtainarigorousproofofanintuitivefact,itsolidifiedthelink intheirmindsbetweentheabstractmanipulationsofformalmathematicsandtheir informalintuitionofmathematics(andoftherealworld),ofteninaverysatisfying way. By the time they were assigned the task of giving the infamous “epsilon and delta”proofsinrealanalysis,theyhadalreadyhadsomuchexperiencewithformal- izing intuition, and in discerning the subtleties of mathematical logic (such as the distinction between the “for all” quantifier and the “there exists” quantifier), that thetransitiontotheseproofswasfairlysmooth,andwewereabletocovermaterial both thoroughly and rapidly. By the tenth week, we had caught up with the non- honoursclass,andthestudentswereverifyingthechangeofvariablesformulafor Riemann–Stieltjesintegrals,andshowingthatpiecewisecontinuousfunctionswere Riemann integrable. By the conclusion of the sequence in the twentieth week, we had covered (both in lecture and in homework) the convergence theory of Taylor PrefacetotheFirstEdition ix andFourierseries,theinverseandimplicitfunctiontheoremforcontinuouslydiffer- entiablefunctionsofseveralvariables,andestablishedthedominatedconvergence theoremfortheLebesgueintegral. Inordertocoverthismuchmaterial,manyofthekeyfoundationalresultswere lefttothestudenttoproveashomework;indeed,thiswasanessentialaspectofthe course,asitensuredthestudentstrulyappreciatedtheconceptsastheywerebeing introduced.Thisformathasbeenretainedinthistext;themajorityoftheexercises consist of proving lemmas, propositions and theorems in the main text. Indeed, I wouldstronglyrecommendthatonedoasmanyoftheseexercisesaspossible—and thisincludesthoseexercisesproving“obvious”statements—ifonewishestousethis texttolearnrealanalysis;thisisnotasubjectwhosesubtletiesareeasilyappreciated justfrompassivereading.Mostofthechaptersectionshaveanumberofexercises, whicharelistedattheendofthesection. To the expert mathematician, the pace of this book may seem somewhat slow, especiallyinearlychapters,asthereisaheavyemphasisonrigour(exceptforthose discussionsexplicitlymarked“Informal”),andjustifyingmanystepsthatwouldordi- narilybequicklypassedoverasbeingself-evident.Thefirstfewchaptersdevelop(in painfuldetail)manyofthe“obvious”propertiesofthestandardnumbersystems,for instancethatthesumoftwopositiverealnumbersisagainpositive(Exercise5.4.1), orthatgiven anytwodistinctrealnumbers,onecanfindrationalnumber between them(Exercise5.4.5).Inthesefoundationalchapters,thereisalsoanemphasison non-circularity—notusinglater,moreadvancedresultstoproveearlier,moreprim- itiveones.Inparticular,theusuallawsofalgebraarenotuseduntiltheyarederived (andtheyhavetobederivedseparatelyforthenaturalnumbers,integers,rationals, andreals).Thereasonforthisisthatitallowsthestudentstolearntheartofabstract reasoning,deducingtruefactsfromalimitedsetofassumptions,inthefriendlyand intuitivesettingofnumbersystems;thepayoffforthispracticecomeslater,whenone hastoutilizethesametypeofreasoningtechniquestograpplewithmoreadvanced concepts(e.g.,theLebesgueintegral). Thetexthereevolvedfrommylecturenotesonthesubject,andthusisverymuch orientedtowardsapedagogical perspective;muchofthekeymaterialiscontained insideexercises,andinmanycasesIhavechosentogivealengthyandtedious,but instructive,proofinsteadofaslickabstractproof.Inmoreadvancedtextbooks,the studentwillseeshorterandmoreconceptuallycoherenttreatmentsofthismaterial, andwithmoreemphasisonintuitionthanonrigour;however,Ifeelitisimportantto knowhowtodoanalysisrigorouslyand“byhand”first,inordertotrulyappreciate the more modern, intuitive and abstract approach to analysis that one uses at the graduatelevelandbeyond. Theexpositioninthisbookheavilyemphasizesrigourandformalism;however thisdoesnotnecessarilymeanthatlecturesbasedonthisbookhavetoproceedthe same way. Indeed, in my own teaching I have used the lecture time to present the intuitionbehindtheconcepts(drawingmanyinformalpicturesandgivingexamples), thus providing a complementary viewpoint to the formal presentation in the text. The exercises assigned ashomework provide anessentialbridge between thetwo, requiring the student to combine both intuition and formal understanding together x PrefacetotheFirstEdition inordertolocatecorrectproofsforaproblem.ThisIfoundtobethemostdifficult task for the students, as it requires the subject to be genuinely learnt, rather than merelymemorizedorvaguelyabsorbed.Nevertheless,thefeedbackIreceivedfrom thestudentswasthatthehomework,whileverydemandingforthisreason,wasalso very rewarding, as it allowed them to connect the rather abstract manipulations of formalmathematicswiththeirinnateintuitiononsuchbasicconceptsasnumbers, sets,andfunctions.Ofcourse,theaidofagoodteachingassistantisinvaluablein achievingthisconnection. Withregardtoexaminationsforacoursebasedonthistext,Iwouldrecommend eitheranopen-book,open-notesexaminationwithproblemssimilartotheexercises given in the text (but perhaps shorter, with no unusual trickery involved), or else a take-home examination that involves problems comparable to the more intricate exercisesinthetext.Thesubjectmatteristoovasttoforcethestudentstomemorize thedefinitionsandtheorems,soIwouldnotrecommendaclosed-bookexamination, oranexaminationbasedonregurgitatingextractsfromthebook.(Indeed,inmyown examinations I gave a supplemental sheet listing the key definitions and theorems whichwererelevanttotheexaminationproblems.)Makingtheexaminationssimilar tothehomeworkassignedinthecoursewillalsohelpmotivatethestudentstowork through and understand their homework problems as thoroughly as possible (as opposedto,say,usingflashcardsorothersuchdevicestomemorizematerial),which isgoodpreparationnotonlyforexaminationsbutfordoingmathematicsingeneral. Someofthematerialinthistextbookissomewhatperipheraltothemaintheme and may be omitted for reasons of time constraints. For instance, as set theory is notasfundamentaltoanalysisasarethenumbersystems,thechaptersonsettheory (Chapters3,8)canbecoveredmorequicklyandwithsubstantiallylessrigour,orbe givenasreadingassignments.Theappendicesonlogicandthedecimalsystemare intendedasoptionalorsupplementalreadingandwouldprobablynotbecoveredin the main course lectures; the appendix on logic is particularly suitable for reading concurrently with the first few chapters. Also, Chapter 5 (on Fourier series) is not neededelsewhereinthetextandcanbeomitted. For reasons of length, this textbook has been split into two volumes. The first volumeisslightlylonger,butcanbecoveredinaboutthirtylecturesiftheperipheral materialisomittedorabridged.Thesecondvolumerefersattimestothefirst,butcan alsobetaughttostudentswhohavehadafirstcourseinanalysisfromothersources. Italsotakesaboutthirtylecturestocover. Iamdeeplyindebtedtomystudents,whoovertheprogressionoftherealanal- ysis course corrected several errors in the lectures notes from which this text is derived, and gave other valuable feedback. I am also very grateful to the many anonymous referees who made several corrections and suggested many impor- tant improvements to the text. I also thank Adam, James Ameril, Quentin Batista, Biswaranjan Behara, José Antonio Lara Benítez, Dingjun Bian, Petrus Bianchi, Phillip Blagoveschensky, Tai-Danae Bradley, Brian, Eduardo Buscicchio, Carlos, cebismellim,MatheusSilvaCosta,GonzalesCastilloCristhian,Ck,WilliamDeng, Kevin Doran, Lorenzo Dragani, EO, Florian, Gyao Gamm, Evangelos Georgiadis, AdityaGhosh,ElieGoudout,TiGong,UlrichGroh,GökhanGüçlü,YaverGulusoy,
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