泛函分析-孙炯版答案--第四章-泛函分析第四章答案

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ݎfSK51.X´Banach˜m,G´X4f˜m,T´dGk.ê˜mmk.‚5Žf,KT˜½Œ±òÿXmk.‚5Žf~T,…÷vk~Tk=kTk:2.x1;x2;


;xn´(X;k
k)¥‚5Ã',f1;2;


;ngK;ùpK“Lê.y²3Xþ3‚5¼f,÷v(1)f(xk)=k;k=1;


;n;(2)kfkM¿‡^‡´:é?¿ê1;2;


;n2K;kjnXk=1kkjMknXk=1kxkk:3.X´‚5D‰˜m,f´Xþšk.‚5¼,K3x02X;¦f(x0)6=0;X=Nfx0g;ùp´¢£½E¤ê,N´f˜m.4.X‚5D‰˜m,x;y2X,eé8f2X,kf(x)=f(y):y²x=y.5.y²1p+1ž,(lp)=lq(1p+1q=1);=f2(lp)…=3a=faig2lq;¦f(x)=1Xi=1aixi;8x=fxig2lp;…kfk=kak:6.aqu15K,y²(l1)=l1;(C0)=l1;Ù¥C0´l1¥ÂñuêN|¤f˜m.7.Á¦e½Â3lpþ‚5Žf
ݎf:(1)Tfx1;x2;


g=f0;x1;x2;


g;(2)Tfx1;x2;


g=f1x1;2x2;


g,Ù¥fkg´k.ê;(3)Tfx1;x2;


g=fx1;x2;


;xn;0;


g,Ù¥n´‰½;(4)Tfx1;x2;


g=fnxn;n+1xn+1;


g,Ù¥fkg´k.ê,n´‰½.8.Á¦e3L2(1;1)þ½Â‚5Žf
ݎf:(1)(Tx)(t)=x(t+h)(h´‰½¢ê);(2)(Tx)(t)=a(t)x(t+h)(a(t)´k.Œÿ¼ê,h´‰½¢ê);(3)(Tx)(t)=12[x(t)+x(t)]:9.L´ll2l2‚5Žf,=(y1;y2;


)=L(x1;x2;


),Ù¥yn=x1+


+xnn2:1
2
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ݎfy²L´k.‚5Žf…kLk6(1Pn=11n2)12,¿¦ÑL.10.T:l2!l2,T(x1;x2;


)=(x1;x22;


;xnn;


):¦T.11.X‚5D‰˜m,y²µXÃ‘˜mž,X´Ã‘˜m.12.X‚5D‰˜m,‚5ŽfA:X!X;D(A)=X;‚5ŽfB:X!X;D(B)=X:XJ(Bf)(x)=f(Ax);8x2X;f2X;y²A;BÑ´k.‚5Žf.13.eTn;T2B(X;Y)(n=1;2;


),y²kTnTk!0ž,kTnTk!0(n!1).14.X,Y´D‰‚5˜m,y²eYØ´…X6=f0g,KB(X;Y)Ø.15.f(t)´[a;b]þŒÿ¼ê,XJéu?˜g(t)2Lq[a;b](1q+1),kf(t)g(t)2L(a;b),Kf(t)2Lp[a;b](1q+1ž,1p+1q=1;q=1ž,p=+1;q=+1ž,p=1).16.X´Hilbert˜m,-fk(x)=(x;yk);k=1;2;x;yk2X;3X¥½Â(f1;f2)=(y1;y2):y²X´Hilbert˜m.17.'(x;y)´Hilbert˜mHþ˜‡
ÝV‚5¼ê,÷v(1)9M0;¦j'(x;y)jMkxkkyk;(2)90;¦j'(x;y)jkxk2:y²é8f2H;3˜ƒyf2H,¦'(x;yf)=f(x);8x2H;¿…yf6uf.18.H´Hilbert˜m,¿3H¥xn!x0;ynw!y0,y²(xn;yn)!(x0;y0):19.L´Hilbert˜mHHþk.‚5Žf,y²L´å…=L´å.20.(H1;(
;
)1);(H2;(
;
)2)´Hilbert˜m,L:H1!H2´k.‚5Žf.½ÂL:H2!H1;(y;Lx)2=(Ly;x)1;8x2H1;y2H2.y²(1)L´k.‚5Žf;(2)L=L;(3)kLk=kLk.21.L´Hilbert˜mHþk.‚5Žf.y²e'XªN(L)=N(LL);R(L)=R(LL):
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22.T´Hilbert˜mH¥g
ݎf…kk._Žf,y²T1´g
ݎf.23.A´ESȘmHþk.‚5Žf,XJéz‡x2H;(Ax;x)=0,KA=0.éu¢˜m,d(J´Ä¤áºXJA´g
ݎf,KØØH´¢½E,‡(Ax;x)=0(8x2H),KA=0.24.T:L2[0;1]!L2[0;1]d(Tx)(t)=tx(t)½Â,y²T´g
Ý.25.T:l2!l2d(1;2;3;


)!(0;0;3;4;


)½Â,Á¯Tk.í?´g
Ýí?26.Tk.‚5Žf,…kTk1,y²fx:Tx=xg=fx:Tx=xg:27.HEHilbert˜m,THþk.‚5Žf,e阃x2H;Re(Tx;x)=0;KT=T:28.T´Œ©Hilbert˜mUþk.‚5Žf,fengU¥IOX.eé?Ûm;n;k(Ten;em)=(Tem;en),KT´g
ݎf.29.XBanach˜m,ffigX:y²é?Ûx2X;P1i=1jfi(x)j+1¿‡^‡´é?ÛF2X;P1i=1jF(fi)j+1:30.fxkg´Banach˜mX¥:.y²XJéuz˜‡f2X;P1k=1jf(xk)j1;K3~êM;¦éuz˜‡f2X;1Xk=1jf(xk)jMkfk:31.X;Y´Banach˜m,T´XY‚5Žf;q8f2Y;x!f(Tx)´Xþk.‚5¼.y²T´ëY.y²fxngX;xn!x2X;Txn!y2Y:Ϗéu8f2X;f(Tx)´Xþ‚5k.¼,¤±f(Txn)!f(Tx);f(Txn)!f(y)(n!1):d4˜5·‚kf(Tx)=f(y)(8f2Y).2ŠâHahn-Banach½níTx=y;lT´4.d4㔽nTëY.32.X´Banach˜m,p(x)´Xþ¼,÷v(1)p(x)0;(2)0ž;p(x)=p(x);(3)p(x+y)p(x)+p(y);¿…x;xn2X;xn!x(n!1)ž,limn!1p(xn)p(x):y²3~êM,¦p(x)Mkxk(x2X):33.y²g‡Banach˜mX´Œ©¿‡^‡´X´Œ©.34.y²?Ûk‘D‰‚5˜mÑ´g‡.
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ݎf35.y²˜mL1[a;b]9l1Ø´g‡.36.xn=f(n)kg2lp(n=1;2;3;


),KfxngfÂñufkg2lp¿‡^‡´supn1kxnk+1,…éz‡k;limn!1(n)k=k.37.y²l1¥?ÛfÂñ:7´rÂñ.38.X´D‰‚5˜m,MX4‚5f˜m.y²µXJfxngM;¿…n!1žx0=wlimn!1xn;Kx02M:39.Tn´Lp(R)(1p1)g²£Žf:(Tnf)(x)=f(x+n);8f2Lp(R);n=1;2;


:y²Tnw!0;´kTnfkp=kfkp;8f2Lp(R):40.HHilbert˜m,x0;xn2H(n=1;2;


);n!1ž,xnw!x0;…kxnk!kx0k.y²xn!x0(n!1):41.HHilbert˜m,fxng´H¥8,Ke^‡dµ(1)P1n=1xnÂñ;(2)P1n=1(xn;y)(8y2H)Âñ;(3)P1n=1kxnk2Âñ.
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1ÊÙSK{‰1.y²Tx=fng2m;x2G:-fn:fn(x)=n;x2G;n=1;2;





:´yfn´Gþk.‚5¼…kfnkGkTk.KdHahn{Banach½n,éz‡n3Xþk.‚5¼Fn¦é8x2GkFn(x)=fn(x)…kFnk=kfnkG.-eT:X!m;eTx=fFn(x)g;x2X:´eT´Xþm‚5Žf.Ϗé8x2XkkeTxksupn1kFnkkxkkTkkxk;eT´k.…keTkkTk:,¡,keTk=supx2X;kxk=1keTxksupx2G;kxk=1kTxk=kTk:KkeTk=kTk:2.y²7‡5.e÷v¤`^‡f3,Ké?¿ê1;2;


;n2K;kjnXk=1kkj=jnXk=1kf(xk)j=jf(nXk=1kxk)j6kfkknXk=1kxkk6MknXk=1kxkk:¿©5.E=spanfx1;x2;


;xng:½Âf0(x)=Pnk=1kk;8x=Pnk=1kxk2E;AOkf0(xk)=k:Kkjf0(x)j=jnXk=1kkj6Mkxk)kf0k6M;8x2E:2ŠâHahn-Banach½n,3Xþ3‚5¼f,¦é8x2Ef(x)=f0(x);…kfk=kf0k6M:3.y²Ïf´Xþšk.‚5¼,K3x02X;x06=0;¦f(x0)6=0.-M=N+fx0g;w,kXM;…N\fx0g=;:
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1ÊÙ
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ݎfe¡yXM.é8x2X,-y=xf(x)f(x0)x0,kf(y)=0,Ky2N.8x2XŒL«x=y+f(x)f(x0)x0;Ù¥y2N;f(x)f(x0)x02fx0g:ùL²XM.KX=Nfx0g:4.y²-x0=xy,Kd^‡é8f2X,kf(x0)=f(x)f(y):díØ5.1.5x=y:5.y²7‡5.é8f2(lp);-ek=(k‡z}|{0;


;0;1;0;0;


)2lp;Ké8x2lp;Œ±L«x=1Xk=1xkek:dfëY5kf(x)=1Xk=1f(ek)xk=1Xk=1kxk;Ù¥k=f(ek):e¡y²=fkg1k=12lq:¯¢þ,é8n2N;Eᑕþx(n);ًIx(n)k=8:jkjq1eik;k6n;0;kn;Ù¥k=arg(k):Ϗéuz˜‡½n;x(n)‹Ikk‘؏§kx(n)2lp.duf(x)=1Xk=1f(ek)x(n)k=1Xk=1kjkjq1eik=nXk=1jkjq;…jf(x)j6kfkkx(n)k=kfk(nXk=1jkj(q1)p)1p=kfk(nXk=1jkjq)1p;KnXk=1jkjq6kfk(nXk=1jkjq)1p)(nXk=1jkjq)1q6kfk;(5.0.1)l=fkg2lq;…kkq6kfk:¿©5.éu82lq;½Âf(x)=1Xk=1kxk;8x=(xk)2lp:
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w,f(x)´lpþ˜‡‚5¼…ŠâHolderت,·‚kjf(x)j1Xk=1jkjjxkjkkqkxkp:(5.0.2)f2(lp).d(5.0.1),(5.0.2)kfk=kkq.6.y²(1)é8f2(l1);-ek=(k‡z}|{0;


;0;1;0;0;


)2l1;Ké8x2l1;Œ±L«x=1Xk=1xkek:dfëY5Œf(x)=1Xk=1f(ek)xk=1Xk=1kxk;Ù¥k=f(ek):Ϗjkj=jf(ek)j6kfkkekk=kfk(k=1;2;


);¤±=fkg2l1:,˜¡,éu8=fkg2l1;½Âf(x)=1Xk=1kxk;(8x=(xk)2l1):w,f(x)´l1þ˜‡‚5¼…jf(x)j61Xk=1jkjjxkj6supk1jkj1Xk=1jxkj=kk