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1、CHAPTER13TheDualityTheoremLetXbealinearspaceoverthereals,dimX=n.ItsdualX'consistsofalllinearfunctionsonX.IfXisrepresentedbycolumnvectorsxofncomponentsx1,...,xn,thenelementsofX'aretraditionallyrepresentedasrowvectorswithncomponentst,...,in.ThevalueofatxisIx1+...+Snxn.(1)Ifw
2、eregardasa1xnmatrixandregardxasannx1matrix,(1)istheirmatrixproductlx.LetYbeasubspaceofX;inChapter2wehavedefinedtheannihilatorY1ofYasthesetofalllinearfunctionst;thatvanishonY,thatis,satisfyty=0forallyinY.(2)AccordingtoTheorem3ofChapter2,thedualofX'isXitself,andaccordingtoTh
3、eorem5there,theannihilatorofY1isYitself.Inwords:iftx=OforallinY1,thenxbelongstoY.SupposeYisdefinedasthelinearspacespannedbymgivenvectorsyi,...,yninX.Thatis,Yconsistsofallvectorsyoftheform(3)Clearly,l;belongstoY1iffYj=0,j=1,...,M.(4)LinearAlgebraandItsApplications,SecondEdi
4、tion,byPeterD.LaxCopyrightQ2007JohnWiley&Sons,Inc.202THEDUALITYTHEOREM203SoforthespaceYdefinedby(3),thedualitycriterionstatedabovecanbeformulatedasfollows:avectorycanbewrittenasalinearcombination(3)ofingivenvectorsyjiffeverythatsatisfies(4)alsosatisfiesty=0.Weareaskingnowf
5、oracriterionthatavectorybethelinearcombinationofmgivenvectorsyjwithnonnegativecoefficients:y=>PjYi,Pi?0.(5)Theorem1(Farkas-Minkowski).Avectorycanbewrittenasalinearcombinationofgivenvectorsyjwithnonnegativecoefficientsasin(5)iffeverythatsatisfiesyj?0,j=(6)alsosatisfies(6)'t
6、y?0.Proof.Thenecessityofcondition(6)'isevidentuponmultiplying(5)ontheleftbyi.ToshowthesufficiencyweconsiderthesetKofallpointsyofform(5).Clearly,thisisaconvexset;weclaimitisclosed.Toseethiswefirstnotethatanyvectorywhichmayberepresentedinform(5)mayberepresentedsoinvariousway
7、s.Amongalltheserepresentationsthereisbylocalcompactnessone,orseveral,forwhichEpjisassmallaspossible.Wecallsucharepresentationofyaminimalrepresentation.Nowlet{z,,}beasequenceofpointsofKconvergingtothelimitzintheEuclideannorm.Representeachzminimally:Z.=Epn.jyj.(5)'Weclaimtha
8、tEp,,,j=Pisaboundedsequence.ForsupposeonthecontrarythatP->oo.Sincethesequencezisconvergen
