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1、FunctionalAnalysis1DrA.J.WassermannLentterm19991ALTXedbyTimPerutz{commentstosoc-archim-notes@lists.cam.ac.uk.EContents1Introduction42Metricspaces62.1Preliminaries...........................62.2Compactnessandcompleteness.................93Normedspaces133.1Continuitypropertiesofthealgeb
2、raicoperations........143.2Completeness...........................143.3Theuniformnorm........................1423.4Thespace`............................15p3.5The`spaces...........................16p3.6TheLspaces...........................193.7Equivalentnorms.........................20
3、3.8Boundedoperatorsandlinearfunctionals............214Innerproductspaces254.1Closestpointsandtheprojectiontheorem...........264.2OrthonormalbasesandParseval'sequation...........304.3OrthonormalbasesandtheGram-Schmidtprocess.......324.4ThedualofaHilbertspace:theRieszrepresentationtheor
4、em................335FourierseriesandtheDirichletproblem365.1TheDirichletproblem......................405.2UniformlyconvergentFourierseries...............426Applicationtotheta,gammaandzetafunctions456.1Thetafunctions..........................456.2Thegammaandbetafunctions..............
5、...476.3TheRiemannzetafunction...................482CONTENTS37TheStone-WeierstrassTheorem507.1Importantexample:Hermitefunctions.............577.2Productsandeasy`Fubini'theorem...............58Note.The nalchapterofthecourse,onLebesgueintegration,isomittedhereasthiswassuppliedinprintedf
6、ormbyDrWassermann.ThesenotesdonotalwaysreproduceverbatimwhatDrWassermannwroteontheboard,butthechangesareminor-theysuitedme,andtheymightnotpleaseeveryone.Iwouldalsoencourageyounottoseethesenotesasasubstituteforgoingtothelectures,whichcontainmorejokesandarebilledbythelectureras`mellow'.
7、..T.P.Chapter1IntroductiontherelectroniclecturenotesareavailablefromtheArchimedeans.Alistofavailablecourses,andthenotesmaybedownloadedfromhttp://www.cam.ac.uk/CambUniv/Societies/archim/notes.htmoryoucanemailsoc-archim-notes@lists.cam.ac.uktogetacopyofthesetsyourequire.4Copyright(c)The
8、Archi
