第二章-赋范线性空间2

94XYBanachA:XY,xyX∈,Rλµ∈()AxyAxAyλµλµ+=+AXY(linearoperator)MA:XY||||||||AxMx≤xX∈AXY(,)BXY(,)LXY(,)BXX()BX(,)ABXY∈{}0||||||||sup||||xXAxAx∈−=||||AA||||1||||1||||sup||||sup||||xxAAxAx≤===2.1()ijAa=mn×:CCnmA→j2jmj111,,,TnnnjjjjjjAxaaaξξξ===⎛⎞=⎜⎟⎝⎠∑∑∑()12,,,CTnnxξξξ=∈nCmCAFrobenius2.2(,)kst[a,b][a,b]:[,][,]ACabCab→()()(,)()baAxtkstxsds=∫()[,]xtCab∈A||||max|(,)|max|()|max|(,)|||||bbaaatbatbatbAxktsdsxtktsdsx∞∞≤≤≤≤≤≤≤=∫∫2.3l∞1()ihhl=∈9y=Hxniniiyhx∞−=−∞=∑:ll∞∞→H||||max||||max||||||||iniininnniiiyhxhxhx∞∞∞∞−∞=−∞=−∞=−∞=≤≤∑∑∑.2.41[,]Cab[a,b]Banach[,]Cab1:[,][,]DCabCab→()()'()Dxtxt=1()[,]xtCab∈D()nntaxtba−⎛⎞=⎜⎟−⎝⎠1n≥||||1nx∞=1()()'()/()nnntaDxtxtnbaba−−⎛⎞==−⎜⎟−⎝⎠||||()nnDxnba∞=→∞→∞−D2.5Fredholm(,)htτ()xt()yt()(,)()baythtxdτττ=∫[a,b](,)htτ2[,][,]|(,)|ababhtdtdττ×+∞∫∫22:[,][,]FLabLab→,()()(,)()baFxthtxdτττ=∫,2()[,]xtLab∈1/22[,][,]|||||(,)|||||ababFxhtdtdxττ×⎛⎞≤⎜⎟⎜⎟⎝⎠∫∫FFredholmXYBanach,(,)ABBXY∈,xX∈,Cλ∈,(),()ABxAxBxAxAxλλ+=+=9AB+(,)ABXYλ∈(,)BXYBanach:IXX→,Ixx=,xX∈:XY→0,xYθ=∈0,xX∈(,)BXY(,)ABXY∈(,)BBYZ∈:()()BABAxBAx=xX∈()ABX∈Nn∈ntimesnAAAA=0AI=1||||||||||||AxAx≤(,)ABXY∈xX∈2,()ABBX∈||||||||||||ABAB≤XYBanach{}|(,)TBXYλλ∈Λ⊂xX∈{}|||||Txλλ∈Λ{}|Tλλ∈Λ(,)BXYM0≥λ∈Λ||||TMλ≤5X:RfX→fXXBanachX*Xf,|()|||||=sup||||xXxfxfxθ∈≠*X,*fgX∈()()()fgxfxgx+=+()()fxfxλλ=xX∈R(orC)λ∈1Rn9*(R)nf∈Rn{}12,,,neeei(),1,2,,ifeinα==.Rnx∈,1niiixeξ==∑,11()()nniiiiiifxfeξαξ====∑∑,Holder1/21/22211|()|||||nniiiifxαξ==⎛⎞⎛⎞≤⎜⎟⎜⎟⎝⎠⎝⎠∑∑,1/221||||||niifα=⎛⎞≤⎜⎟⎝⎠∑.1nfiiixeα==∑,221()||||||nfififxxα===∑,1/221||||||niifα=⎛⎞≥⎜⎟⎝⎠∑.1/221||||||niifα=⎛⎞=⎜⎟⎝⎠∑.12(,,,)nfααα→.,12(,,,)Rnnααα∈,1Rnniiixeα==∈∑,1()niiifxαξ==∑,1/21/211|()|||||nniiiifxαξ==⎛⎞⎛⎞≤⎜⎟⎜⎟⎝⎠⎝⎠∑∑,fRn,1/221||||||niifα=⎛⎞=⎜⎟⎝⎠∑.()*12(,,,)Rnnfααα→∈.*(R)nf∈,12(,,,)Rnnααα∈1()niiifxαξ==∑,1Rnniiixeα==∈∑,1/221||||||niifα=⎛⎞=⎜⎟⎝⎠∑.,()*RRnn=21ll∞,()*1ll∞=.()*1fl∈,12(,,)fxlαα∞=∈1()iiifxαξ∞==∑,112(,,)xlξξ=∈,||||||||ffx∞=.31p∞,()*pqll=,111,1,qpq+=p,q.41p∞,()*L(X)L(X)pq=,XR[a,b],111,1,qpq+=9p,q.:.:Hahn-Banach.:*fX∈,0f≠,{}M|()fxXfxαα=∈=,1)0fMX;2)0xX∈0()0fx≠,00{|R}fXMxλλ=+∈;3)0()fxα=,00ffMxMα=+.fMαX.(Eidelhei):0G,1GE,o01GG=∅∩,H0G1G,*fX∈,Rα∈10sup()inf()xGxGfxrfx∈∈≤≤6TXX→BanachX(0,1)α∈xyX∈||-||||-||TxTyxyα≤TXTXX→BanachX*xX∈**=Txx*xT()=0xϕ*x*x-()Txxxϕ→Txx||||1||||Txxα≤9BanachXBanachTXX→T,xyX∀∈||||||||01TxTyxyαα−≤−0xX∈0=nnxTx=12n{}=1nnx∞Cauchy1n≥+1-1-1||-||=||-||||-||nnnnnnxxTxTxxxα≤+110||-||||-||nnnxxxxα≤1mn≥()-1-2-1-1-2+110||-||||-||+||-||++||-||+++||-||mmnmnmmmmnnxxxxxxxxxxααα≤≤-10101-=||-||||-||1-1-mnnnxxxxααααα01αlim=0nnα→∞lim||-||=0mnmnxx→∞{}=1nnx∞Cauchylim*nnxxX→∞=∈Tlim*nnTxTx→∞=*1()nnTxxxn+=→→∞**Txx=*xT*x**xT*********||||||||||||TxTxxxxxα−=−≤−01α***xx=nnx*x*10()||||||||1nnrnxxxxαα=−≤−−1.TBanachXSSTS2.:TXX→NNTTX2*xNT1*(*)*(*)NNNTxTTxTxTTx+===*TxNT**xTx=1Jacobi=Axb()ija=AnT12(,,,)Rnnbbb=∈bT12x=(,,)nxxxi1||||nijiijjiaa=≠∑=Axbx9111122(,,,)00nnnnaDdiagaaaa⎛⎞⎜⎟==⎜⎟⎜⎟⎝⎠-1-1xx-Dx+DbTA=xRn∈Rn||||∞TTT1212(,,,)(,,,)Rnnnxxxyyy==∈x,y||||||()||||()()||TTT∞∞∞−==-1xyx-yI-DAx-y()ijIa−=-1DA0,,ijijiiijaaija=⎧⎪=⎨−≠⎪⎩11||||1||nnijijjjiijiaaa==≠=∑∑1111111||||max|()|max||||max|max||nnnijjjijjjijiiininininjjjjijijiTTaxyaxyaxy∞≤≤≤≤≤≤≤≤===≠≠≠−=−≤−≤−∑∑∑xy11max||||||nijinjjia∞≤≤=≠⎛⎞⎜⎟=⎜⎟⎜⎟⎝⎠∑x-y11max||1nijinjjiaα≤≤=≠=∑||||||||TTα∞∞−≤xyx-yT=AxbJacobiT0(1,0,,0)=x1,0,1,2,nnTn+==xx*n→xx*x=Axb*10()||||||||1nnrnαα∞∞=−≤−−xxxx2Fredhom()fs[a,b](,)kst[,][,]abab×0M|(,)|baKstdtM∫[,]sab∈1||Mλ()[,]sCabφ∈()()(,)()basfsksttdtφλφ=+∫**()[,]fsCab∈**Fredhom:[,][,]TCabCab→()()()(,)()baTsfsksttdtφλφ=+∫()[,]tCabφ∈912,[,]Cabφφ∈12121212||||max|||(,)||()()||||(,)|max|()()|||||||bbaaasbatbTTkstttdtkstdtttMφφλφφλφφλφφ∞∞≤≤≤≤−≤−≤−≤−∫∫||1Mαλ=12||||TTφφ∞−≤12||||αφφ∞−T*[,]Cabφ∈(*)()()(,)*()baTsfskstsdsφλφ=+∫sin(,)stkststππ=()(,)()basksttdtλφφ=∫ProlaleSpheroidalfunctions.3(,)kst,asbats≤≤≤≤Voterra()()(,)()saxsfskstxtdtλ=+∫***()[,]fsCab∈Rλ∈*()[,]xsCab∈:[,][,]TCabCab→()()(,)()saTxsfsKstxtdtλ=+∫[,]xCab∈(),()[,]xtytCab∈|()()||||(,)(()())|||||||()saTxsTyskstxtytdtMxysaλλ∞−=−≤−−∫[,]sab∈M|(,)|,,kstMasbats≤≤≤≤≤1m11111|()()||||(,)()()||||||||||(,)|()(1)!()||||||!smmmmammsmammmTxsTyskstTxtTytdtMxyksttadtmsaMxymλλλλ−−−−−∞∞⎡⎤−=−⎣⎦−≤−−−≤−∫∫asb≤≤||||max|()()|||||mmmmmasbTxTyTxsTysxyα∞∞≤≤−=−≤−||()!mmmmMbamλα−=lim0mmα→∞=N1NαNT[,]Cab2T*()[,]xsCab∈***9{0,1}X=nYXXX=×××Y(,)Vd(,)(,),1(,)dxydxyxyVdxy=∈+∼(,)Vd∼(),||||XX⋅(),||||YY⋅()()1222::,XYfXYRfxyxy×→=+(),xyXY∈×XY×MHn{}nyyy,,21MHx∈∀Mxy()()nnyyyGyyyxGz,,,,,,21212=zxy=−2[0)L,+∞M123fff,,23123()()()tttfteftefte−−−=,=,=1102()102txtt⎧,,⎪⎪=⎨⎪⎪⎩MM()xt[1,1]−0niiiat=∑n1210()|niiixtatdt−=|−∑∫ia1221/210()|(1)niiixtattdt−−=|−−∑∫ia