linearalgebraanditsapplications duality

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1、CHAPTER2DualityReaderswhoaremeetingtheconceptofanabstractlinearspaceforthefirsttimemaybalkatthenotionofthedualspaceaspilinganabstractionontopofanabstraction.Ihopethattheresultspresentedattheendofthischapterwillconvincesuchskepticsthatthenotionisnotonlynaturalbutusefulf

2、orexpeditiouslyderivinginterestingconcreteresults.Thedualofanonmedlinearspace,presentedinChapter14,isaparticularlyfruitfulidea.Thedualofaninfinite-dimensionalnormedlinearspaceisindispensablefortheirstudy.LetXbealinearspaceoverafieldK.Ascalarvaluedfunction1,I:X-K,define

3、donX,iscalledlinearifl(X+y)=I(x)+1(y)(1)forallx,yinX,andl(kx)=kl(x)forallxinXandallkinK.Notethatthesetwoproperties,appliedrepeatedly,showthatl(ktxt++ktl(xi)++Wedefinethesumoftwofunctionsbypointwiseaddition;thatis,(1+in)(x)=l(x)+m(x).LinearAlgebraandItsApplications.Seco

4、ndEdition,byPeterD.LaxCopyright2007JohnWiley&Sons,Inc.1314LINEARALGEBRAANDITSAPPLICATIONSMultiplicationofafunctionbyascalarisdefinedsimilarly.Itiseasytoverifythatthesumoftwolinearfunctionsislinear,asisthescalarmultipleofone.ThusthesetoflinearfunctionsonalinearspaceXits

5、elfformsalinearspace,calledthedualofXanddenotedbyV.EXAMPLEI.X={continuousfunctionsf(s),0

6、iablefunctionsfon10,1]}.Forsin[0,1],It1(f)_a;aJf(s)isalinearfunction,wherea-denotesthejthderivative.Theorem1.LetXbealinearspaceofdimensionn.TheelementsxofXcanberepresentedasarraysofnscalars:X=(c,,...,cn),(3)Additionandmultiplicationbyascalarisdefinedcomponentwise.Leta,

7、,...,anbeanyarrayofnscalars;thefunctionlbedefinedby(4)isalinearfunctionofx.Conversely,everylinearfunctionIofxcanbesorepresented.Proof.Thatl(x)definedby(4)isalinearfunctionofxisobvious.Theconverseisnotmuchharder.LetIbeanylinearfunctiondefinedonX.Definexjtobethevectorwho

8、sejthcomponentis1,withallothercomponentszero.Thenxdefinedby(3)canbeexpressedasx=c,x,++cnxn.Denote1(x1)byaj;itfollowsf