linear mappings between normed linear spaces

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1、CHAPTER15LinearMappingsBetweenNormedLinearSpacesLetXandYbeapairoffinite-dimensionalnonmedlinearspacesoverthereals;weshalldenotethenorminbothspacesbyII,althoughtheyhavenothingtodowitheachother.Thefirstlemmashowsthateverylinearmapofonenonmedlinearspaceintoanotherisbounded.Lemma1.ForanylinearmapT:X-+Y,

2、thereisaconstantcsuchthatforallxinX,ITxiaixj;(2)thenTx=>ajTxj.BypropertiesofthenorminY,ITxllaillTxllFromthiswededucethatITxi

3、PPLICATIONSwhereIxlx=maxlail,k=ITxil.WehavenotedinChapter14thatIIxisanorm.SincewehaveshowninChapter14,Theorem2,thatallnormsareequivalent,IxI,

4、lideanspaceintoanother.Analogously,wehavethefollowingdefinition.Definition.ThenormofthelinearmapT:X-->Y,denotedasITI,isITxIITI=sup.(4)a-#0IxIRemark1.Itfollowsfrom(1)thatITIisfinite.Remark2.ItiseasytoseethatITIisthesmallestvaluewecanchooseforcininequality(1).Becauseofthehomogeneityofnorms,definition(

5、4)canbephrasedasfollows:(4)'ITI=supITxI.jxl=1Theorem2.ITIasdefinedin(4)and(4)'isanorminthelinearspaceofalllinearmappingsofXintoY.Proof.SupposeTisnonzero;thatmeansthatforsomevectorxoy60,Tx000.Thenby(4),IT*ITI>IxolsincethenormsinXandYarepositive,thepositivityofITIfollows.LINEARMAPPINGSBETWEENNORMEDLIN

6、EARSPACES231Toprovesubadditivitywenote,using(4)',thatwhenSandTaretwomappingsofX-rY,thenIT+SI=supI(T+S)xI

7、eityisobvious;thiscompletestheproofofTheorem2.GivenanymappingTfromonelinearspaceXintoanotherY,weexplainedinChapter3thatthereisanothermap,calledthetransposeofTanddenotedasT,mappingY,thedualofY,intoX',t