Basic Real Analysis - Stony Brook University

1 Basic real AnalysisDigitalSecondEditionsBy Anthony W. KnappBasic AlgebraAdvanced AlgebraBasic real Analysis ,with an appendix ElementaryComplex Analysis Advanced real AnalysisAnthony W. KnappBasic real AnalysisWith an Appendix Elementary Complex Analysis Along with a Companion VolumeAdvanced real AnalysisDigital Second Edition, 2016 Published by the AuthorEast Setauket, New , aknappTitle:BasicRealAnalysis,withanappe ndix ElementaryComplexAnalysis (2010):28 01,26 01,42 01,54 01,34 01,30 01,32 ,ISBN-13978-0-8176-3250-2c" auserBostonDigitalSecondEdition,nottobes old,noISBNc" ,images,andotherdatacontainedinthisfile, whichisinportabledocumentformat(PDF),are proprietarytotheauthor,andtheauthorretai nsallrights,includingcopyright, ,trademarks,servicemarks,andsimilaritems ,eveniftheyarenotidentifiedassuch, userBoston,c/oSpringerScience+BusinessMe diaInc.,233 SpringStreet,NewYork,NY10013,USA, ,scholarship,andresearch,andforthesepurp osesonly, ,postitonline,andtransmititdigitallyforp urposesofeducation,scholarship, ( ,EPUB),theymaynoteditit, ,usersmustchargenofee, ,noextractsorquotationsfromthisfilemaybe usedthatdonotconsistofwholepagesunlesspe rmissionhasbeengrantedbytheauthor(andbyB irkh userBostonifappropriate).

2 Thepermissiongrantedforuseofthewholefile andtheprohibitionagainstchargingfeesexte ndtoanypartialfilethatcontainsonlywholep agesfromthisfile, (andbyBirkh userBostonifappropriate).Inquiriesconcer ningprintcopiesofeithereditionshouldbedi rectedtoSpringerScience+ SusanandTo My Children, Sarah and William,andTo My real -AnalysisTeachers:SalomonBochner, WilliamFeller, HillelFurstenberg,Harish-Chandra, Sigurdur Helgason,John Kemeny,John Lamperti,HazletonMirkil,Edward Nelson,LaurieSnell,Elias Stein,Richard ,Sequences, (S) , ,L2,L , ,SingularMeasures, , , Arzel`aandStone ,Matrices, , ,VI,andVIII, ,II, THESECONDEDITIONI ntheyearssincepublicationofthefirstediti onofBasicRealAnalysis, manyread-ers have reactedto the bookby sendingcomments,suggestions,and overall comprehensive natureof the book,associatingthisfeaturein part with the large numberof problemsthat develop so many sidelightsand applicationsof the theory.

3 Somepeoplewonderedwhethera way mightbe foundfor a revisionto includesomeminimaltreatmentof Stokes s Theoremand complex Analysis ,despitethe reservationsI expressedin the generalcommentsand specificsuggestionswerecorrections,well over a hundredin all, that neededto be addressedin any ofthe correctionswereof minormatters,yet readersshouldnot have to copewitherrorsalongwithnew resultsin the first editionneededto be deletedor seriouslymodified,and additionalresultsand the first edition,the authorgranteda publishinglicenseto Birkh auserBostonthat was limitedto printmedia,leavingthe questionof electronicpubli-cationunresolved. A majorchangewith the secondeditionis that the questionofelectronicpublicationhas now beenresolved, and a PDFfile, calledthe digitalsecondedition, maybe downloadedfromthe author s own Web pageand mainchangesto the first editionofBasicRealAnalysisare as follows: A carefultreatmentof arc length,line integrals,and Green s Theoremforthe planehas beenaddedat the end of ChapterIII.

4 Theseaspectsof Stokes sTheoremcan be handledby the samekindsof techniquesof real analysisas in the first aspectsof Stokes s Theoremin higherdimensionswouldrequirea greatdeal moregeometry, for reasonsgiven ,and that moregeneraltreatmenthas notbeenincluded. The core of a first coursein complex analysishas beenincludedas on thoseaspectsof elementarycomplex analysisthat are usefulas toolsin real appendixincludesmorethan 80 problems, ChaptersI IIIas a the appendixfits into the plan of the bookis explainedin the Guidefor the the SecondEdition A new sectionin ChapterIX proves and appliesthe Riesz ThorinConvexityTheorem,a fundamentalresultaboutLpspacesthat takes advantageof ele-mentarycomplex Analysis . About20problemshavebeenaddedattheendsofC haptersI of threekinds:someillustratethe new topicsof arc length,line integrals,and Green s Theorem;somemake use of elementarycomplex analysisasin AppendixB to shedfurtherlighton resultsand problemsin the variouschapters;and somerelateto the topicof Banachspacesin ChapterXII.

5 The correctionssent by readersand by reviewershave mostsignificantsuchcorrectionwas a revisionto the proofof Zorn s Lemma,theearlierproofhavinghad a materialin AppendixB is designedas the text of part of a first courseincomplex taughtsucha coursemyselfon one courseincomplex analysisinvariablybegins withsomepreliminarymaterial,and that canbetakenfromChaptersItoIII;detailsappe arintheGuidetotheReader. AppendixB formsthe coreof the course,dealingwithresultshavingan analyticflavor,includingthe part of the theorydue to Cauchy. The topicof conformalmapping,whichhas a moregeometricflavor, has beenomitted,and someinstructorsmightfeel obligedto includesomethingon this topicin the statesthe RiemannMappingTheoremat one pointbut doesnot prove it; all the toolsneededfor its proof,however, are presentin the appendixand its instructorwill end a first coursein complex analysiswithmaterialon infiniteseriesand productsof functions,or of aspectsof the theoryof specialfunctions,or on any was BenjaminLevitt, Birkh ausermathematicseditorin New York, whoencouragedthe writingof this secondedition,whomadea numberof suggestionsaboutpursuingit, and whopassedalongcommentsfromseveral anonymousrefereesaboutthe strengthsand weaknessesof the am especiallygratefulto thosereaderswhohave sentme commentsover the of thecorrectionsthat weremadewerekindlysent to me eitherby S.

6 H. Kimof kindlysent by GlennJia of longcorrectionto the proofofZorn s Lemmaresultedfroma discussionwithQiu typesettingwasdoneby the programTexturesusingAMS-TEX, and the figuresweredrawn as withthe first edition,I invite correctionsand as longas I am able,I planto pointto a list of known correctionsfrommy own homepage, W. KNAPPF ebruary2016 PREFACETO THEFIRSTEDITIONThis book and its companionvolume,AdvancedReal Analysis , systematicallydevelop conceptsand tools in real analysisthat are vital to every mathematician,whetherpureor applied,aspiringor bookstogethercontainwhat the young mathematicianneeds to know about real analysisin order tocommunicatewell with colleaguesin all branchesof ,andtheirprimaryaudienceisstudentswhoare learningthematerialforthefirsttimeandwho areplanninga careerinwhichthey will use advanced mathematicsprofessionally. Much of the materialin thebooks correspondsto normalcourse work.

7 Nevertheless,it is often the case thatcoremathematicscurricula,time-limite dastheyare, donotincludeallthetopicsthatonemightlike . Thusthebookincludesimportanttopicsthatma ybeskippedin requiredcoursesbut that the professionalmathematicianwill ultimatelywantto learn by contentof the requiredcoursesat each University reflectsexpectationsofwhatstudentsneedbe forebeginningspecializedstudyandworkonat hesis. Theseexpectationsvaryfromcountrytocountr yandfromuniversitytouniversity. Evenso,thereseemstobearoughconsensusabou twhatmathematicsaplenarylecturerat a broad internationalor nationalmeetingmay take as known by the tablesof contentsof the two booksrepresentmy own understandingof whatthat degree of knowledgeis for real topics and featuresofBasicReal Analysisare as follows: Early chapterstreat the fundamentalsof real variables,sequencesand seriesof functions,the theoryof Fourier series for the Riemannintegral, metricspaces,and the theoreticalunderpinningsof multivariablecalculusand ordi-nary differentialequations.

8 Subsequentchaptersdevelop the Lebesguetheory in Euclideanand abstractspaces,Fourier series and the Fourier transformfor the Lebesgueintegral,point-settopology, measuretheory in locallycompactHausdorff spaces,andthe basics of Hilbertand Banachspaces. The subjectsof Fourier series andharmonicfunctionsare used as recurringmotivation for a numberof theoreticaldevelopments. Thedevelopmentproceedsfromtheparticulart othegeneral,oftenintroducingexampleswell before a theory that to the First Edition More than 300 problemsat the ends of chaptersilluminateaspectsof thetext, develop relatedtopics,and point to separate55-pagesection HintsforSolutionsofProblems attheendofthebookgivesdetailedhints for most of the problems,togetherwith a standardcalculussequencein one and several variables,the mostimportantprerequisitefor usingBasicReal Analysisis that the reader alreadyknow what a proof is, how to read a proof, and how to write a obtainedfrom honorscalculuscourses,or from a courseinlinearalgebra, ,itisassumedthatthereaderiscomfortablewi thamodestamountoflinearalgebra,including row reductionof matrices,vector spaces and bases, and the associatedgeometry.

9 A passingacquaintancewith the notionsof group,subgroup,andquotientis helpfulas IV are appropriatefor a single rigorousreal-variablescourse andmay be used in either of two ways. For studentswho have learnedabout proofsfromhonorscalculusorlinearalgebra, thesechaptersofferafulltreatmentofrealva riables,leaving out only the more familiarparts near the beginning suchaselementarymanipulationswith limits, convergence tests for infiniteseries withpositive scalarterms, and routinefacts aboutcontinuityand differentiability. Forstudentswho have learnedabout proofs from a first junior-seniorcourse in realvariables,thesechaptersareappropriat eforasecondsuchcoursethatbeginswithRiema nnintegrationand sequencesand series of functions;in this case the firstsectionof ChapterI will be a review of some of the more difficult foundationaltheorems,and the coursecan concludewith an introductionto the Lebesgueintegral from ChapterV if time throughXII treat Lebesgueintegrationin various settings,as wellas introductionsto the EuclideanFourier transformand to materialistaught at the graduatelevel in the UnitedStates,fre-quentlyinoneofthreeways : ThefirstwaydoesLebesgueintegrationinEucl ideanand abstractsettingsand goes on to considerthe EuclideanFourier transforminsome detail; this correspondsto ChaptersV VIII.

10 A secondway does LebesgueintegrationinEuclideanandabstrac tsettings,treatsLpspacesandintegrationon locallycompactHausdorff spaces,and concludeswith an introductionto Hilbertand Banachspaces;this correspondsto ChaptersV VII, part of IX, and XI transformwith some of the subjectof partial differentialequations;thiscorrespondsto some portionof ChaptersV VI and VIII, followed by chaptersfrom the companionvolume,AdvancedReal my own teaching,I have most often built one course aroundChaptersI IVand anotheraroundChaptersV VII, part of IX, and XI XII. I have normallyPreface to the First Editionxviiassignedthe easier sectionsof ChaptersII and X as outsidereading,indicatingthe date when the lectureswould begin to use that the bookmaybe usedwithcoursesmaybe deducedfrom the chart DependenceamongChapters on page xiv and thesection Guideto the Reader on pages xv problemsat the ends of chaptersare an importantpart of the book.