; TeX output 2000.03.19:2023ҳ3-L͡7L͍63׉ffiYff'pnH DU ffUFfd͠UffH~- cmcsc10DefinitionTwl}h! cmsl12NormonaLinearSpaceNormedSpaceRw[OFunctionalAnalUTysisUUffUffUFḧ́ffffffiY\63׉ffiYff'pnH DU ffUFfd͠UffH~Definition[wXInnerProSductYw[OFunctionalAnalUTysisUUffUffUFḧ́ffffffiY63׉ffiYff'pnH DU ffUFfd͠UffH~Definition[wV:LinearTVransformation/OpSeratorYw[OFunctionalAnalUTysisUUffUffUFḧ́ffffffiY*ҳ3-L͡7L͍63׉ffiYff'pnH DcffUFfd͠Dք:ffH*K`y cmr10A7real-vqalued7function !", cmsy10jj b> cmmi10xjjde nedonalinearspaceX,=wherex2X,=is7said toUUbGea': cmti10normonX7if"V cmbx10P9ositivity8,jjxjj0,T riangleTInequalit9ygjjx8+y[ٸjjjjxjj8+jjyjj,Homogeneit9yJ.#jj zxjj=j j8jjxjj,UU ^ϲanarbitraryscalar,P9ositiveTDe nitenesspzjjxjj=0UUifandonlyifx=0,whereUUxandy.arearbitrarypGointsinX.AUUlinear/vectorspacewithanormiscalledanorme}'dspace.U:ff<ffUFḧ́ffffffiY\63׉ffiYff'pnI DcFffUFfd͠9sffH,LetuĵX>bGeacomplexlinearspace.Aninnerpr}'oducthonuĵXisamappingthat assoGciates2toeachpairofvectorsx,*y) ascalar,denoted(x;y[ٲ),thatsatis estheUUfollowingpropGerties:Additivit9y;ѵ(x8+y[;zp)=(x;zp)8+(y[;zp),Homogeneit9yJ.#( BZx;y[ٲ)= z(x;y[ٲ),Symmetry<,a(x;y[ٲ)=Lщfe2/(y;x),P9ositiveTDe nitenesspz(x;x)>0,UUwhenx6=0.U9sff9uffUFḧ́ffffffiY63׉ffiYff'pnH DcffUFfd͠Dք:ffH*AtransformationLof(opGeratoron)alinearspaceX_intoalinearspaceY8, wherew͵X@andYhavewthesamescalar eld,issaidtobGealine}'artransformation(op}'erator)if81.L( zx)= L(x);8x2X7andUU8scalars ,and82.L(xٓRcmr71S+8x2|s)=L(x1)8+L(x2)UUforallx1;x2C2X.U:ff<ffUFḧ́ffffffiY;ҳ3r"V cmbx10': cmti10}h! cmsl12- cmcsc10 !", cmsy10 b> cmmi10K`y cmr10ٓRcmr7 p