《最新泛函分析讲义(英文版)》麻省理工学院

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1、FuntionalAnalysisLecturenotesfor18.102RichardMelroseDepartmentofMathematics,MassachusettsInstituteofTechnol-ogyE-mailaddress:rbm@math.mit.eduVersion0.8D;Revised:4-5-2010;Run:May4,2010.ContentsPreface5Introduction6Chapter1.NormedandBanachspaces91.Vectorspaces92.Normeds

2、paces103.Banachspaces124.Operatorsandfunctionals145.Completion166.Baire'stheorem207.Uniformboundedness218.Openmappingtheorem219.Closedgraphtheorem2310.Hahn-Banachtheorem2411.Doubledual2712.Problems2813.SolutionstoProblems31Chapter2.TheLebesgueintegral431.Stepfunctions

3、442.Lebesgueintegrablefunctions463.Coveringlemmas474.Setsofmeasurezero485.Monotonicitylemmas506.LinearityofL1(R)527.Theintegraliswell-dened52R8.Theseminormjfj569.Nullfunctions5810.ThespaceL1(R)6011.Thethreeintegrationtheorems6112.ThespacesLp(R)6413.Lebesguemeasure671

4、4.Problems6815.SolutionstoProblems73Chapter3.Hilbertspaces891.pre-Hilbertspaces892.Hilbertspaces903.Orthonormalsets914.Gramm-Schmidtprocedure9134CONTENTS5.Completeorthonormalbases926.Isomorphismtol2937.Parallelogramlaw938.Convexsetsandlengthminimizer949.Orthocomplemen

5、tsandprojections9510.Riesz'theorem9611.Adjointsofboundedoperators9712.Compactnessandequi-smalltails9813.Finiterankoperators10014.Compactoperators10215.Weakconvergence10316.ThealgebraB(H)10617.Spectrumofanoperator10818.Spectraltheoremforcompactself-adjointoperators1081

6、9.FunctionalCalculus11120.Compactperturbationsoftheidentity11221.Fredholmoperators11522.Problems11623.Solutionstoproblems129Chapter4.Applications1431.FourierseriesandL2(0;2):1432.Dirichletproblemonaninterval1463.Harmonicoscillator1504.CompletenessofHermitebasis1535.F

7、ouriertransform157Appendix:ExamPreparationProblems161Bibliography165PREFACE5PrefaceTheseareinformalnotesfor18.102fromandforcoursesatMITinSpring2009and2010.Useatyourowncaution!6CONTENTSIntroductionWhatisitallfor?Mainaims:-Tobeabletocarryout`standard'constructionsin(lin

8、ear)functionalanalysis:AbstractHilbertspace{oneineachdimensionConcreteHilbertspace{Many,suchasL2([0;1])Exampleofatheorem:-Th