Mathematical Reviews for Microeconomics

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1、MathematicalReviewsforMicroeconomicsAdamChiLeungWongShanghaiUniversityofFinanceandEconomicsFall2009Thisnoteprovidesanon-exhaustivelistofmathematicalconceptsandthe-oremsthatwewilluseinouradvancedmicroeconomicscourse.Throughoutthisnote,Isupposeyouknowthebasicconceptsandbasicoperationsofre

2、alnumbers,vectors,matrices,andsets.References:ChapterA1andSubsectionA2.1in:JehleandReny(JR),AdvancedMicro-economicTheory,2ndedition,ShanghaiUniversityofFinanceandEconomicsPress,2003.1EuclideanspacesLetRdenotethesetofrealnumbers.LetR+denotethesetofnon-negativerealnumbers,andR++thesetofpo

3、sitiverealnumbers.Thatis,R+=fx2R:x0gR++=fx2R:x>0g:Letnbeapositiveinteger.ThenRnisthen-dimensionalspaceofrealnumbers,i.e.Rn=f(x;x;


;x):x2R;


;x2Rg:12n1nRnandRnare:+++Rn=f(x;x;


;x)2Rn:x0;


;x0g+12n1nRn=f(x;x;


;x)2Rn:x>0;


;x>0g:++12n1nForanytwopointsx=(x;


;x)andy=(y;


;y)in

4、Rn(orRn,or1n1n+Rn),theEuclideandistanceisafunctiond:RnRn!Rde…nedby:++q22d(x;y)=(x1y1)+


+(xnyn):1Remark:1.d(x;y)0foranyx;y2Rn.2.d(x;y)=0ifandonlyifx=yforanyx;y2Rn.3.d(x;y)=d(y;x)foranyx;y2Rn.4.Triangleinequality:d(x;y)d(x;z)+d(z;y)foranyx;y;z2Rn.Then-dimensionalEuclideanspaceisthe

5、setRnthatusetheEuclideandistancetomeasurethedistanceofanytwopointsinit.Throughoutthisnote,"distance"alwaysmeanstheEuclideandistance,andIwillonlyconsidertheEuclideanspace.2ConvexsetsDe…nition:SRnisaconvexsetifforallx12Sandx22S,wehavetx1+(1t)x22Sforalltintheinterval0t1.Theorem:LetSand

6、TbeconvexsetsinRn.ThenSTisaconvexset.(Veryeasytoprove!)3SequencesDe…nition:AsequenceinRnisalistofelementsx;x;x;


2Rn.Itis1231customarytowriteasequenceintheformoffxigi=0orfxig.De…nition:WesayasequencefxginRnconvergestoapointx2Rnifixntendstoxasntendsto1,i.e.forallrealnumber>0,thereisan

7、integerN(dependingon)suchthat:nN)d(x;xn)<:Wecanalternativelysayfxngisconvergenttox,orsayxisthelimitofthesequencefxngandwrite:limxn=xorxn!x:n!1Asequencecanhaveatmostonelimit.(Thatis,ifalimitexists,itisunique.)Asequenceissaidtobeconvergentifithassomelimit.Asequenceissaidtob