泛函分析习题及参考答案

2R21212(,),(,)xxxyyyR==∈2211122(,)()()dxyxyxy=−+−21122(,)max{,}dxyxyxy=−−31122(,)dxyxyxy=−+−2122ddd≤≤31322ddd≤≤2322ddd≤≤),(yxdX),(1),(),(~xydxydxyd+=X,0),(~≥yxdyxyxdyxd=⇔=⇔=0),(0),(~),(~),(1),(),(1),(),(~yxdyxdyxdxydxydxyd=+=+=ttt+−=+1111),(),(),(yzdzxdyxd+≤),(),(1),(),(),(1),(),(),(1),(),(),(1),(),(~yzdzxdyzdyzdzxdzxdyzdzxdyzdzxdyxdyxdyxd+++++=+++≤+=),(~),(~),(1),(),(1),(yzdzxdyzdyzdzxdzxd+=+++≤1p≥1()()(,,,)innpnxlξξ=∈,2,1=n1(,,,)pixlξξ=∈n→∞1()1(,)0ppnniiidxxξξ∞=⎛⎞=−→⎜⎟⎝⎠∑)1(n→∞()niiξξ→1,2,i=)2(0ε∀0N()1pnpiiNξε∞=+∑n1()1(,)0ppnniiidxxξξ∞=⎛⎞=−→⎜⎟⎝⎠∑()niiξξ→1,2,i=1(,,,)pixlξξ=∈0ε∀10N11()2ppiiNεξ∞=+∑1nN()1()2pnpiiiεξξ∞=−∑11111()()111ppppppnnpiiiiiNiNiNξξξξε∞∞∞=+=+=+⎛⎞⎛⎞⎛⎞⎜⎟≤−+⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠∑∑∑1nN11,2,nN=20N2()1pnpiiNξε∞=+∑{}12max,NNN=()1pnpiiNξε∞=+∑n0ε∀0K()1()2pnpiiKεξ∞=+∑n1()2ppiiKεξ∞=+∑()niiξξ→0NNn()1Kpnpiiiξξε=−∑()()()111(,)KppppnnnniiiiiiiiiKdxxξξξξξξ∞∞===+=−=−+−∑∑∑11()()111()()2pKpppnnpppiiiiiiKiKξξξξε∞∞==+=+⎛⎞≤−++⎜⎟⎝⎠∑∑∑],[baLp)1(≥p()1(,)()()bppadxyxtytdt=−∫)(txn)(tx)1()(txn)(tx)2({})(txn],[ba0),→xxnρ0∀σ∫∫≥−−≥−)()()(σxxEpnEpnndtxxdttxtx)((σσ≥−⋅≥xxEmnp,2,1=n∞→n0)((→≥−σxxEmn)(txn)(tx)(tx0ε01δEe⊂1δme∫eppdttx2))((1ε0ε0NNn∫−Eppndttxtx2)()(1εε+−≤+−≤∫∫∫∫∫peppEpnpeppepnpepndtxdtxxdtxdtxxdttx11111)()()()())(ε∫pepndttx1))(,1,+=NNnNn,,2,1=02δEe⊂2δme∫epndttxε)(),min(21δδδ=Ee⊂δmeε∫pepndttx1))(n{})(txn],[ba0ε)()(εε≥−=xxEEnn0)(→εnmE0δ0NNnδε)(nmE)()(εε−=xxEFnn∫∫−+−=nnEFpnpnnpdtxxdtxxxx),(ρpEppEpEpnpnnnndtxdtxdtxx∫∫∫⎥⎥⎦⎤⎢⎢⎣⎡+≤−11)()(pFpnabdtxxnε⋅−−∫)(0ε0δEe⊂δme2))(1ε∫pepndttx2))(1ε∫pepdttx0δ0NNnδε)(nmEpEpnndtxx∫≤−ε0ε0NNn1(,)(1)pndxxbaε≤+−⋅)(txn)(txBX,21nOOOB∩∞==1nnOB,2,1,1==nnnδ∪BxnnxO∈=)(δnOB⊂nO),2,1(=n∩∞=∈1nnOxBxn∈nxxdn1),(),2,1(=nxxn→BBx∈∩∞==1nnOBX21FFX21GGΦ=21GG∩11FG⊃22FG⊃Φ=21FF∩0),(2111∈∀FxdFx0),(,1222∈∀FxdFx2),(,2),(122211FxdFxd==δδ∪∪2211)(,)(222111FxFxxGxG∈∈==δδ,21GG21FF210GGx∩∈211220110,,),(,),(FxFxxxdxxd∈∈δδ),(2),(2),(),(),(),(211221200121xxdFxdFxdxxdxxdxxd≤++≤Φ=21GG∩∞l(){},1,0,,,,21=∈=∞nnlMξξξξ{}{}Myxnn∈==ηξ,yx≠,)sup1nndxyξη=−=M∞l{}ny∞⊂∈lMx1(,)3Oxny1(,)3Ox1(,)3OyMyx∈,11(,)(,)33OxOy=Φ∩1(,)3OxxM⎧⎫∈⎨⎬⎩⎭{}ny{}ny1,3nOylM∞⊃⊃∪M1(,)3nOyMyx,(,)1dxy=2,),)(,)3nndxydxydyy≤+XAX),(inf)(yxdxfAy∈=Xx∈)(xfXXx∈0Xx∈),(),(),(),(inf)(00xydxxdyxdyxdxfAy+≤≤=∈,Ay∈)(),()(00xfxxdxf+≤)(),()(00xfxxdxf+≤),()()(00xxdxfxf≤−)(xf0x)(xfXTXYYFFT1−XFY{}1nxTF−⊂nxx→Xx∈nTxF∈TnTxTx→TxF∈1xTF−∈Xx∈{}nxX⊂nxx→nTxTx→00ε{}knxX⊂0(,)kndTxTxε≥{}0(,)FydyTxε=≥{}knTxF⊂FYFT1−X1knxTF−∈,knxx→1xTF−∈TxF∈00(,)0dTxTxε=≥nTxTx→pl)1(≥p{}pnxl⊂1()()()2(,,,,)innnnxξξξ=,2,1=n0ε∀0N,mnN()()1pnmpiiiξξε∞=−∑1,2,i={}()niξRRiRξ∈()niiξξ→n→∞s()()1spnmpiiiξξε=−∑m→∞()1spnpiiiξξε=−≤∑()1pnpiiiξξε∞=−≤∑12(,,,,)ixξξξ=111()()111ippppppnniiiiiiξξξξ∞∞∞===⎧⎫⎧⎫⎧⎫≤+−⎨⎬⎨⎬⎨⎬⎩⎭⎩⎭⎩⎭∑∑∑pxl∈(,)ndxxε≤nxx→pl)1(≥p[,]Cab111111,2(),1,2nnnnnntxtnttt−−≤≤−⎧⎪=−≤≤⎨⎪≤≤+⎩,2,1=n{}[,]nxCab⊂nm2211(,)()()nmnmdxxxtxtdtmn−=−=−∫{}nx([,],)Cabd{}nx([,],)Cabd()[,]xtCab∈22(,)()()0nndxxxtxtdt−=−→∫11221()1()0nnxtdtxtdt−−++−→∫∫(0)1x−=−(0)1x+=()[,]xtCab∈{}nx([,],)Cabd[,]Cab)(xfR1)(≤′αxfxxf=)(Ryx∈,yxyxfyfxf−⋅≤−⋅′=−αξ)()()(10αfRxxf=)(FnnRAF)(,yxFyx≠∈),(),(yxdAyAxdAF)(inf),,()(0xAxxdxFxϕϕϕ∈==)(xϕFFx∈•)(0•=xϕϕ0),()(=•••Axxdxϕ),(),(2••••AxxdxAAxd0)()(ϕϕϕ=••xAx)(inf0xFAxFxϕϕ∈•=∈0),(=••Axxd••=xAxAF•≠=xxxAx000,Fx∈0),(),(),(000xxdAxAxdxxd•••=AF1p≥12(,,,,)pnxlξξξ=∈limppxx∞→+∞=X1x2xba,212xbxxa⋅≤≤⋅Xx∈0b21xbx⋅≤nXxn∈21nnxnx⋅nxxnn121111=nnxx1x2x0bXx∈21xbx⋅≤0′a12xax⋅′≤aa′=10aXx∈12xxa≤⋅,2,1,0,0,,=≠≠∈nxxXxxnn∞→→nxxn,xxxxnn→xxn→xxxxnn−≤−,xxn→xxxxxxxxxxxxxxxxxxnnnnnnnn⋅−+−⋅⋅⋅=⋅⋅−⋅=−)()(1∞→→−⋅+−⋅⋅⋅≤nxxxxxxxxnnn,0)(1xxxxnn→21XXBanach{},,,,21nxxxx=nnXx∈,2,1=n∑∞=∞1npnxXXpnpnxx11⎟⎠⎞⎜⎝⎛=∑∞=1≥pXBanachX,2,1),,,,,()()(2)(1)(==ixxxxiniiiXIjiI∃∀,,0,0εε−)()(jixx∑∞=−11)()()(nppjninxxεnε−)()(jninxx{}∞=1)(iinxnXnX{}nnXx⊂∞→i,2,1,)(=→nxxnin),,,,(21nxxxx=∑∞=−=−11)()()(nppninixxxx∑∑∞=∞=∞→−=−11)(11)()()()(limnppninnppjninjxxxxIiε≤−xxi)(∞→→ixxi,)(∑∑∞=∞=−≤=11)(11)()(nppninnppnxxxx++∞∑∞=11)()(nppinxXx∈X()()nnijnijRxxxxnjiRaaAAxzzTx∈⋅⋅⋅=≤≤∈===,,,,,1,,,,21nR{}1maxiinxx∞≤≤=11niixx==∑12221()niixx==∑TnnRR→TT∞1T2T111111()()nnnijjijjijinininjjjTxmaxaxmaxaxmaxax∞∞≤≤≤≤≤≤====≤⋅≤⋅∑∑∑T11nijinjTmaxa∞≤≤=≤∑1110nnijmjinjjMmaxaa≤≤====≠∑∑1mn≤≤012((),(),,())Tmmmnxsignasignasigna=01x∞=01111()()nnnijmjmjmjmjinjjjTxmaxasignaasignaaM∞≤≤====≥==∑∑∑TM∞≥11nijinjTmaxa∞≤≤==∑1111111111()()()nnnnnnnijjijjjijijjnijijjiiTxaxaxxamaxax≤≤========≤⋅=≤∑∑∑∑∑∑∑T111nijjniTmaxa≤≤=≤∑1110nnijimjniiMmaxaa≤≤====≠∑∑1mn≤≤0xme011x=011nimiTxaM===∑TM∞≥111nijjniTmaxa≤≤==∑0A≠TAA()0TmaxAAλ()()()()()11122222()()TTTTmaxTxAxAxxAAxAAxλ==≤0nxR∈021x=()1202()TmaxTxAAλ=T()122()TmaxTAAλ=]11[−C∫∫−−=0110)()()(dttxdttxxf∫∫∫∫∫−−−=+≤−=0110110110)()()()()()(dttxdttxdttxdttxdttxxf]1,1[,2)(max211−∈∀⋅=⋅≤≤≤−Cxxtxt2≤f⎪⎪⎪⎩⎪⎪⎪⎨⎧−−−−−=]1,1[,1]1,1[,]1,1[,1)(nnnntntxn,2,1,1],1,1[==−∈nxCxnn∫∫∫∫=−=−−−−−+=−−−nnnnnnndtdtntdtntdtxf10111101,2,1,12)1()()(1)(,2,1,12)(=−=≥nnxffn2≥f2=f()ija(,2,1,=ji)∞∑∞=≥11supjijia∞l∞l),,(21xxx=),,(21yyy=yTx=∑∞=⋅=1jjijixay,2,1=iT∞l∞l∑∞=≥=11supjijiaTT∞∈=lxn),,,,(21ξξξ∑∑∑∞=≥∞=≥∞=≥≥⋅≤⋅≤⋅==1111111supsupsupsupjijijjijijjijiiiaxaaTxξξη∑∞=≥≤1