泛函分析-孙炯版答案--第五章

1oÙk.‚5ŽfSK41.supn1janj1,3l1þ½ÂŽfT:y=Tx;Ù¥x=fkg;y=fkg;k=kk(k=1;2;


):y²T´l1þk.‚5Žf¿…kTk=supn1janj.2.G´D‰˜mXf˜m,x02X;y²x02G…=éuXþ?˜÷vf(x)=0(x2G)k.‚5¼f7kf(x0)=0:3.X‚5D‰˜m,f´Xþ‚5¼.y²(1)fëY¿‡^‡´f˜mN(f)=fxjf(x)=0g´X¥4f˜m;(2)f6=0ž,fØëY¿‡^‡´N(f)3X¥È—.4.T´C[a;b]þk.‚5Žf,PTtn=fn(t);n=0;1;2;


;y²Td¼êffn(t)g˜(½.5.X;Y;ZÑ´Banach˜m,eT12B(X;Z);T22B(Y;Z),…é8x2X;Žf§T1x=T2yk˜)y=Tx;y²T2B(X;Y).6.X;YBanach˜m,T2B(X;Y),eT´÷Úü,y²3~êa;b;¦é8x2X;kakxkkTxkbkxk:7.MnL«nn¢Ý˜m,éuA=(aij)2Mn,½Ân(A)=Pi;jjaijj.(1)y²kAxk16n(A)kxk1,ùpx2Rn,Rnäk‰êkxk1=nPi=1jxij;(2)A;B2M:y²n(AB)6n(A)n(B).8.(
)´½Â3[a;b]þ¼ê.-(Tx)(t)=(t)x(t)(x2C[a;b]);KT´dC[a;b]Ùgk.‚5Žf¿‡^‡´(
)3[a;b]þëY.9.éuz‡2L1[a;b];½Â‚5ŽfT:Lp[a;b]!Lp[a;b];(Tx)(t)=(t)x(t);8x2Lp[a;b]:¦T‰ê.10.ĎfT:C1[1;1]!C[1;1]µ(Tx)(t)=dx(t)dt;8x2C1[1;1]:1
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1oÙk.‚5ŽfùpC1[1;1]´3[1;1]¥˜êëYN¼ê.(1)eC1[1;1]¥‰ê´kxk1=maxfmax16t61jx(t)j;max16t61jx0(t)jg:¯T´Äk.;(2)eC1[1;1]¥‰ê´kxk2=max16t61jx(t)j;¯T´Äk..11.éf2L[a;b];½Â(Tf)(x)=Zxaf(t)dt:y²(1)eTL[a;b]!C[a;b]Žf,KkTk=1;(2)eTL[a;b]!L[a;b]Žf,KkTk=ba:12.3C[0;1]þ½Â‚5¼f(x)=Z120x(t)dtZ112x(t)dt:y²(1)f´ëY;(2)kfk=1;(3)Ø3x2C[0;1];kxk61;f(x)=1.13.x(t)2C[a;b];f(x)=x(a)x(b);y²f´C[a;b]þk.‚5¼,¿¦kfk:14.¦¼f(x)=R10ptx(t2)dt3±eü«œ/e‰êkfk:(1)x(t)2C[0;1];(2)x(t)2L2[0;1]:15.(t)2C[0;1];3C[0;1]þ½Â¼(f)=Z10(t)f(t)dt;8f2C[0;1];¦kk.16.é?Ûf2L[a;b],Š(Tf)(x)=Rxaf(t)dt.rTÀL[a;b]!C[a;b]Žfž,Áy²kTk=1.17.éuz‡2C[a;b],½Â‚5ŽfT:L1[a;b]!L1[a;b],(Tx)(t)=(t)x(t),y²kTk=kk,Ù¥kkL«3C[a;b]¥‰ê.18.áÝA=(ajk)÷vM=supj1Xk=1jajkj1:
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éu8x=fx1;x2;


g;y=fy1;y2;


g2l1;½Â‚5ŽfT:l1!l1;x!y;Ù¥yj=P1k=1ajkxk;j=1;2;


,y²kTk=M:19.ÄC[0;1]þŽfSfTng,Ù¥(Tnx)(t)=x(t1+1n),KfTngrÂñu,˜k.‚5Žf,ØU‰êÂñuTŽf.20.Tn´Lp(R)(1p1)gŽf.(Tnf)(x)=8:f(x);jxjn;0;jxjn:Ù¥f2Lp(R):y²TnrÂñuðŽfI,ؘ—ÂñI:21.X;Y´D‰‚5˜m,Ln(n=1;2;


)´lXYëY‚5Žf,b½L´lX–YN,¿…é?¿n=1;2;


,3Mn0¦kLxLnxk6Mnkxk;8x2X., Mn!0(n!1).y²L´lXYëY‚5N(K8`²eSfLng˜—Âñ,K§47´ëY!‚5).22.(121K)belimn!1kLnxLxk=0;8x2X,Ò´`LnrÂñuL,KL´‚5.23.E!E1!E2Ñ´Banach˜m,Tn!T2B(E;E1);Sn;S2B(E1;E2),efTng!fSng©OrÂñuT!S,y²fSnTngrÂñuST.24.X;Y´‚5D‰˜m,Tn2B(X;Y);A´¦supn1kTnxk1:xN,K‡oA=X,‡oA´X¥1˜j8.25.ÄS˜mc0=fxx=(x1;


;xn;0;


);8xi2R;n1g;Ù¥z‡ƒx´–õkõ‡Ø0êi¤áS,¿…kxk=supn1jxnj;8x2c0:éuc0þŽfSTm:c0!c0;Tm(x)=(0;


;0;mxm;0;


):(1)OŽkTmk;(2)y²éuz‡x2c0;supm1kTmxk1;(3)y²c0gØ´1j.26.X´ål˜m.F´Xþ¢ëY¼êx…äk5Ÿµéuz˜‡x2X;3~êMx0,¦éuz˜‡F2F;jF(x)jMx:y²3m8U±9~êM0,¦éuz˜‡x2U9¤kF2F:kjF(x)jM:27.X´Banach˜m,X0´X4f˜m.½ÂN:X!X=X0:x![x];8x2X;Ù¥[x]L«¹xûa.y²´mN.28.X´l1¥kkõ‘؏S¤f˜m.½ÂT:X!X;x=(x1;


;xn;


)!y=(y1;


;yn;


);ª¥yk=1kxk;y²
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1oÙk.‚5Žf(1)T2B(X);¿OŽkTk;(2)T1Ã..ù´Ä†Banach_Žf½ngñ?29.ek
k´C[a;b]þ,˜‰ê£‰êPk
k1¤,¿…kxnxk!0ž7kjxn(t)x(t)j!0;8t2[a;b];Kk
k†k
k1d.30.H=L2[0;1];T=iddtD(T)=fu2Hju(0)=0;u3[0;1]þýéëYg:y²T´4Žf.31.X;Y´‚5D‰˜m,D´X‚5f˜m,T:D!Y´‚5N.y²(1)eTëY,D´48,KT´4Žf;(2)eTëY…´4Žf,KY%¹D4.32.X;Y‚5D‰˜m,T:X!Y‚5Žf,eT4Žf…_ŽfT1:Y!X3,y²T1´4Žf.33.X;Y‚5D‰˜m.eT1:X!Y´4Žf…T22B(X;Y),y²T1+T2´4Žf.
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1oÙSK{‰1.y²w,T´l1l1‚5Žf.ϏkTxk=1Xn=1jnnjsupnjnj:kxk;¤±T´k.‚5Žf…kTksupnjnj:,˜¡,en=(0;


;0;1|{z}n ;0;


)2l1;kenk=1;KkTkkTenk=jnj(n=1;2;3;


);¤±kTksupnjnj:nþ,kTk=supn1janj.2.y²7‡5.dux02G;3fxngG¦xn!x0(n!1):f´Xþ?˜÷vf(x)=0(x2G)k.‚5¼.KdfëY5f(x0)=f(limn!1xn)=limn!1f(xn)=0:¿©5.bx0=2G.½Â¼fXe:f(x0+x)=;2C;x2G:w,fX1=fx0+xj2C;x2Ggþ‚5¼.dux0=2G;d(x0;G)6=0;l6=0žkkx0+xk=jjkx0+1jjxkjjd(x0;G):¤±kf(x0+x)k=jj1d(x0;G)kx0+xk;=f´k..´„,f(x)=f(0x0+x)=0(8x2G);f(x0)=1:ù†f(x0)=0gñ.x02G:3.y²(1)7‡5w,.¿©5.N(f)4,·‚y²é80;90;¦kxkž,kjf(x)j.ؔf6=0;x06=0¦f(x0)=: 8ÜN(f)+x0=fx0+x:x2N(f)g:
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1oÙk.‚5ŽfϏN(f)4,¤±N(f)+x04,…062N(f)+x0,l30;¦S(0;)\(N(f)+x0)=;;u´Œ±y²x2S(0;)ž,jf(x)j:¯¢þ,Ø,,=3x2S(0;),¦jf(x)j:-y=xf(x);Kkykkxk;…f(yx0)=f(x0)=0;y2S(0;)\(N(f)+x0);gñ.(2)¿©5.^‡y{.f´ëY,KdN(f)ȗ5Œ,é8x2XŒ3N(f)¥˜Sfxng,¦xn!x(n!1):u´limn!1f(xn)=f(x),=f(x)=0:ù†Kf6=0ƒgñ.7‡5.fØëY,Kd‚5ŽfëY5½n,f73x=0?ØëY.l300†˜‡:fxng2X,¦xn!0ž,jf(xn)j0:é8x2X,w,xf(x)f(xn)xn2N(f),…n!1ž,xf(x)f(xn)xn!x,N(f)3X¥È—.4.y²8x(t)2C[a;b],dWeierstrass½n3fPn(t)=nPi=1iti2P[a;b]g¦kPnxk!0:duT´k.,lëY.Tx(t)=T(limn!1Pn(t))=limn!1T(nXi=1iti)=1Xn=1nfn(t):dué8x(t)2C[a;b]Œdfng1n=1˜(½,¤±Td¼êffn(t)g˜(½.5.y²8x1;x22XéuŽf§T1x1=T2y1;T1x2=T2y2k˜)y1=Tx1;y2=Tx2;T1(x1+x2)=T1x1+T1x2=T2y1+T2y2=T2(y1+y2):ÏdkT(x1+x2)=Tx1+Tx2;ÓnŒT(x1)=T(x1);82K:KT´‚5Žf.e¡y²T´k..¯¢þ,xn!x;Txn!y.duT1xn=T2(Txn)…T1;T2ëY,¤±T1x=T2y,=y=Tx.KT´4,2d4㔽nT´k.,=T2B(X;Y):
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6.y²dK,X;Y´Banach˜m.T2B(X;Y)…T´V,KdBanach_Žf½n,T1´k..Ϗ1=kTT1kkTkkT1k;kT1k0;kTk0:-a=1=kT1k;b=kTk:KŒakxkkTxkbkxk:7.y²(1)éuA=(aij)2Mn;x=(xj)2Rn:Ax=(nXj=1a1jxj;nXj=1a2jxj;


):kAxk1=nXi=1nXj=1aijxjnXi=1nXj=1jaijjjxjjnXi=1nXj=1jaijj
nXj=1jxjj=n(A)kxk1:(2)A=(aij)2Mn;B=(bij)2MnK(AB)ij=nXk=1aikbkjn(AB)=Xi;jnXk=1aikbkjXi;jnXk=1jaikj
nXk=1jbkjj=nXinXk=1jaikjo
nXjnXk=1jbkjjo=n(A)
n(B)8.y²7‡5.eT´dC[a;b]Ùg‚5Žf.x12C[a;b];K(Tx)(t)=(t)2C[a;b]:=(
)3[a;b]þëY.¿©5.e(t)2C[a;b];´T´‚5.dukTx