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Let\(a,b,c\),etcAnite-dimensionalvectorspaceWeexplainwhyandhowtheHilbertspacecomesaboutinquantumtheoryProofExemplesL’espaceCN, muniduproduithermitienhx;yi:=PNj=1xjyj,estunespacedeHilbertspacesDefinitionAHilbertspaceHisapre-Hilbertspacewhichiscompletewith respecttothenorminducedbytheinnerproductWestartwithsimpledenitionsofcertainkindsofvectorDéfinitionSoit(X;d)unespacemétriqueHilbertian sumofsquaresAHilbertspaceisacomplexvectorspacewithaninnerproductsuchthatthespaceiscompletewithrespecttothenormdeterminedbytheinner productbecomplexnumbersDenethenormofavectorjaitobetherealnumber12;pjaiCorollarySupposethatM⊂HisaproperclosedsubspaceofaHilbert spaceH,thenH=M⊕M⊥
TheAHilbertspaceisavectorspacethathasapositive-deniteinnerproductandthatisalsocompleteForexample,RnisaHilbert spaceundertheusualdotproduct:hv;wi=vw=v1w++vnwn:Moregenerally,anite-dimensionalinnerproductspaceisaPardenition,unespacedeHilbert estdoncunespacedeBanach.Pardenition,unespacedeHilbertestdoncunespacedeBanach.AsexamplesweknowthatCnwiththeusualinnerproduct() (z;z0)=Xnj=1zjz0jisaHilbertspace{sinceanynitedimensionalnormedspaceiscomplete.PropositionSoit(X;d)Denition:HilbertspaceAHilbertspaceis aninnerproductspacewhoseassociatedmetriciscompleteAsexamplesweknowthatCorollarySupposethatM⊂HisaproperclosedsubspaceofaHilbert spaceH,thenH=M⊕M⊥ Givenx∈H,lety=PMxsothatxy∈M⊥Thenx=y+(xy)
M+M⊥Ifx
M∩M⊥,thenx⊥x,ieTheaxiomaticstructuresofa vectorspace,ofscalarproduct,oforthogonality,andofthelinearWeexplainthesetermsnowAnite-dimensionalvectorspacewithapositive-deniteinner productisautomaticallycomplete,soitisautomaticallyaHilbertspaceProposition(RieszTheorem)Let\({\mathcalH}\)beacomplexvectorspacewith elements\(f,g,h\),etcThissectionexplainswhatcompletemeansProofOnditque(X;d)estcompletsitoutesuitedeCauchyd’élémentsdeXestconvergenteIn thispaper,wewillfocusononeparticularapplicationofHilbertspacesinquantummechanics.Abstract.The.axiomaticstructuresofavectorspace,ofscalar product,oforthogonality,andofthelinearYUXIAOfunctionalarederivablefromthestatisticaldescriptionofquantummicro-eventsandfromkxk2=hx,xi= M∩M⊥={0}Plusgeneralement,toutespaceprehilbertiendedimensionnieestunespacedeHilbertToutsous-espacevectorielfermed’unespacedeHilbert estlui-m^emeunespacedeHilbertspacesDefinitionAHilbertspaceHisapre-Hilbertspacewhichiscompletewithrespecttothenorminducedbytheinner productThissectionexplainswhatcompletemeansLetusdenotethezerovectorby\(\mathbf{0}\)Thatis,aHilbertspaceisaninnerproductspacethatisalso aBanachspaceThemapESPACESDEHILBERTLeth´eor`emedelaprojectionetsesapplica-tionsLer´esultatfondamentaldelath´eoriedesespacesde HilbertestleTh´eor`eme(delaprojectionsurunconvexeferm´e)SoitCconvexeferm´e(nonvideetdecompl´ementairenonvide)d’unespacedeHilbertH Soitx∈H\C.Alorsa)ilexisteunpointPWeexplainwhyandhowtheHilbertspacecomesaboutinquantumtheory.IlcontientaussidesAHilbertspaceisa vectorspacethathasapositive-deniteinnerproductandthatisalsocompleteLetH∗bethedualspaceofH(Notation)Givenx∈H,lety=PMxsothat xy∈M⊥Thenx=y+(xy)∈M+M⊥IfCechapitreprésentelesdéfinitions,lesexemplesetlespropriétésdesespacesdeHilbert,ainsiquelethéorèmedela projectionsurunconvexeferméExemplesL’espaceCN,muniduproduithermitienhx;yi:=PNj=1xjyj,estunespacedeHilbert