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1、UniversityofMinnesota-DepartmentofEconomicsMathRefresherCourse-Summer2011July28,2011AimoftheCourse:TheaimofthiscourseistointroducesomefundamentalmathematicaltoolsrequiredinMicroeconomicandMacroeconomicTheorysequencesofrstyearinPhD.TentativeOutlineoftheC
2、ourse:0.Howtoreadanddoproofs(halfofalecturetime)1.Realanalysis(1.5weeks)a.Supremumprincipleb.Innerproductspaces,NormedSpaces,MetricSpaces,TopologicalSpacesc.OpenandClosedSets,Interior,ClosureandBoundaryofaSetd.Sequencese.Compactnessf.Continuity,Completen
3、ess2.MathematicalPreliminariesforDynamicProgramming(1week)a.Correspondences,Upperandlowersemi-continuityb.Contractionmappingtheoremc.Theoremofmaximumd.DynamicProgramming3.ConvexAnalysisandOptimization(1week)a.Convexsets,Convexhull,CaratheodaryTheoremb.A
4、nesubspacesofRnc.Separationtheoremsd.Unconstrainedmaximizatione.Constrainedmaximization,Kuhn-TuckerTheoremf.Fixedpointtheorems4.SpecialTopic:IntroductiontoMeasureTheoryorOptimizationinTopologicalVectorSpaces(Iftimepermits)References:0.Solow,D.Howtodoandr
5、eadtheproofs,4thed.,Wiley,2004.1.RudinW.,PrinciplesofMathematicalAnalysis,3rded.,McGraw-Hill,19762.FollandG.B.,RealAnalysis:ModernTechniquesandTheirApplications,2nded.,Wiley,1999.3.StokeyN.andLucasR.withPrescottE.,RecursiveMethodsinEconomicDynamics,Harva
6、rdUniversityPress,1989.4.Mas-ColellA.,WhinstonM.D.,andGreenJ.R.MicroeconomicTheoryOxfordUniversityPress,1995.1NumberTypesAnintegerisanywholenumber,negativeorpositive.Forexample,56or-56isaninteger.pArationalnumberisonethatcanbeexpressedintheformwherepqand
7、qarebothintegersandq6=0.Forexample,−4isarationalnumbersinceitcanbeexpressedas−4.Anotherrationalnumbercouldbe9.5whichcanbe1expressedas19.Anirrationalnumberisonethatcannotbeexpressedin2√theaboveform.Probablythemostcitedirrationalnumbersareπand2.8、hespaceofrealnumbers.Arealnumbercanbeabsolutelyanynumberthatisnotacomplexnumber.Thusarealcanbeadecimalorawholenumber,anirrationalorarationalnumber;negativeorpositive.Inlinearalgebrathereisextensiveuseofvectorsindifferent<−d