theory of functions of a real variable - s. sternberg

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1、Theoryoffunctionsofarealvariable.ShlomoSternbergMay10,20052Introduction.Ihavetaughtthebeginninggraduatecourseinrealvariablesandfunctionalanalysisthreetimesinthelastfiveyears,andthisbookistheresult.Thecourseassumesthatthestudenthasseenthebasicsofrealvariabletheoryandpointsettopol

2、ogy.Theelementsofthetopologyofmetricsspacesarepresented(inthenatureofarapidreview)inChapterI.Thecourseitselfconsistsoftwoparts:1)measuretheoryandintegration,and2)Hilbertspacetheory,especiallythespectraltheoremanditsapplications.InChapterIIIdothebasicsofHilbertspacetheory,i.e.wh

3、atIcandowithoutmeasuretheoryortheLebesgueintegral.Theherohere(andperhapsforthefirsthalfofthecourse)istheRieszrepresentationtheorem.Includedisthespectraltheoremforcompactself-adjointoperatorsandapplicationsofthistheoremtoellipticpartialdifferentialequations.Thepdematerialfollowscl

4、oselythetreatmentbyBersandSchecterinPartialDifferentialEquationsbyBers,JohnandSchecterAMS(1964)ChapterIIIisarapidpresentationofthebasicsabouttheFouriertransform.ChapterIVisconcernedwithmeasuretheory.ThefirstpartfollowsCaratheodory’sclassicalpresentation.ThesecondpartdealingwithHa

5、usdorffmeasureanddi-mension,Hutchinson’stheoremandfractalsistakeninlargepartfromthebookbyEdgar,Measuretheory,Topology,andFractalGeometrySpringer(1991).Thisbookcontainsmanymoredetailsandbeautifulexamplesandpictures.ChapterVisastandardtreatmentoftheLebesgueintegral.ChaptersVI,andV

6、IIIdealwithabstractmeasuretheoryandintegration.ThesechaptersbasicallyfollowthetreatmentbyLoomisinhisAbstractHar-monicAnalysis.ChapterVIIdevelopsthetheoryofWienermeasureandBrownianmotionfollowingaclassicalpaperbyEdNelsonpublishedintheJournalofMathemat-icalPhysicsin1964.Thenwestu

7、dytheideaofageneralizedrandomprocessasintroducedbyGelfandandVilenkin,butfromapointofviewtaughttousbyDanStroock.Therestofthebookisdevotedtothespectraltheorem.Wepresentthreeproofsofthistheorem.Thefirst,whichiscurrentlythemostpopular,derivesthetheoremfromtheGelfandrepresentationthe

8、oremforBanachalgebras.ThisispresentedinChapterIX(forbo