泛函分析-孙炯版答案-第三章

1nÙSȘm†Hilbert˜mSK31.Þіü‡‚5D‰˜mX,¦3Xþ‰êØUdSÈ)¤.2.fxngSȘmH¥:.kxnk!kxk,…(xn;y)!(x;y)(y2H)(n!1),y²:xn!x(n!1).3.En´n‘‚5˜m,fe1;e2;


;eng´En˜‡Ä,(ij)(i;j=1;2;


;n)´½Ý,éEn¥ƒx=Pni=1xiei9y=Pni=1yiei,½Â(x;y)=nXi;j=1ijxiyj;(3.0.1)K(
;
)´Enþ˜‡SÈ.‡ƒ,(
;
)´Enþ˜‡SÈ,K73½Ý(ij)¦(3.0.1)¤á.4.H´SȘm,x1;x2;


;xn´H¥ƒ,§‚÷v^‡(x;x)=8:0;6=;1;=:y²fx1;x2;


;xng´‚5Ã'.5.x;y´ESȘmX¥ü‡šƒ,K(1)kx+yk=kxk+kyk…=y´xê;(2)kxyk=jkxkkykj…=y´xê;(3)‰½z2X;kxyk=kxzk+kzyk…=32[0;1],¦z=x+(1)y:6.ei2X;keik=1(i2N);a2=Pi6=j(ei;ej)21;x=fig2l2;K(1a)kxk22kXieik2(1+a)kxk22:7.XSȘm,x;y2X,b½kx+(1)yk=kxk;8(0661).y²x=y.eX´D‰˜mØ´SȘmž,œ¹qXÛ?8.H´Hilbert˜m,fxngH,÷vP1n=1kxnk1:y²P1n=1xn3H¥Âñ.9.fHng(n=1;2;


)˜SȘm.-H=ffxngjxn2Hn;1Xn=1kxnk21g:1
2
1nÙSȘm†Hilbert˜méufxng;fyng2H:½Âfxng+fyng=fxn+yng(;2K);(fxng;fyng)=1Xn=1(xn;yn):y²H´SȘm,¿…z˜‡HnÑ´Hilbert˜mž,H´Hilbert˜m.10.éuSȘmH,eã^‡dµ(1)x?y;(2)kx+ykkxk;82C;(3)kx+yk=kxyk;82C.11.eSȘmX´¢,Kkx+yk2=kxk2+kyk2%¹Xx?y,eX´E˜mž,x?y™7¤á.Þ~`²ƒ.12.M=fxjx=fxng2l2;x2n=0;n=1;2;


g;y²M´l24f˜m,…¦ÑM?.13.X=Rk,A=fag,Ù¥a=(a1;


;ak),y²A?=f(x1;


;xk)2RkkXj=1ajxj=0g:14.X´SȘm,AX.y²A?=A?.15.XÚY´Hilbert˜mH‚5f˜m,-X+Y=fx+yjx2X;y2Yg.y²(X+Y)?=X?\Y?.16.3L2[a;b]¥,-S=fe2inxg1n=1:(1)ejbaj1.¦yS?=f0g;(2)eba1.¦yS?6=f0g:17.M;N´SȘmHf˜m,M?N,L=MN,y²L´4f˜m¿©7‡^‡´M;Nþ4f˜m(¿©5Ü©b½H).18.3C[1;1]=X¥.-(1)M1=ff2Xjf(x)=0;8x0g;(2)M2=ff2Xjf(0)=0g:OŽM1;M23X¥'uSÈ(f;g)=R11f(x)g(x)dxÖ.19.eH´SȘm,M;NH,K(1)eM?N,KMN?;NM?;(2)M?=(M)?.20.(X;k
k)´D‰˜m.(1)y²D‰˜m(X;k
k)´î‚à,…=,é?¿x;y2X;x6=0;y6=0,
3
kx+yk=kxk+kyk7kx=y(0).(2)y²3î‚àD‰˜m¥,éuz‡x2X,x'u?¿4f˜mYZ%C´˜.21.D‰‚5˜mX¡Š´˜—à,eX¥?Û÷vkxnk=kynk=1;kxn+ynk!2Sfxng;fyngkkxnynk!0.(1)y²?ÛSȘmÑ´˜—à.(2)C[a;b]Ø´˜—à.(3)L1[a;b]Ø´˜—à.22.fx1;x2;x3g´SȘmX¥‚5Ã'8,b½fx1;x2g÷v(xi;xj)=ij;16i;j62:½Âf:C2!RXe:f(1;2)=k1x1+2x2x3k:y²i=(x3;xi);i=1;2ž,fŠ.23.feg(2I)´SȘmH¥IOX.y²éuz‡x2H;x'uù‡IOXFourierXêf(x;e)j2Ig¥õkŒê‡Ø.24.M´H4‚5f˜m,feng†fe0ng©O´M†M?IOÄ,y²feng[fe0ng¤HIOÄ.25.HHilbert˜m,eEH´‚5f˜m¿…éu?¿x2H,x3EþÝK3,KE´4.26.A=fekg´SȘmX¥IOX.y²é8x;y2X,k1Xk=1(x;ek)(y;ek)kxkkyk:27.HHilbert˜m,fekg;fe0kg´H¥ü‡IOX,¿…1Pk=1keke0kk21.y²XJfekg;fe0kg¥ƒ˜´,K,˜‡´.28.Þ~`²SȘm¥IOXؘ½´.29.¡Hn(t)=(1)net2dndtnet2Hermiteõ‘ª,-en(t)=(2nn!p)1=2et22Hn(t);(n=1;2;3;


);y²feng|¤L2(1;+1)¥˜‡IOX.30.-Ln(t).X(Laguerre)¼êetdndtn(tnet),y²f1n!et=2Ln(t)g(n=1;2;


)|¤L2(0;1)¥˜‡IOX.31.y²3Œ©SȘm¥,?˜IOXõ˜Œê8.
4
1nÙSȘm†Hilbert˜m1nÙSK{‰1.)(1)lp(p6=2)þ‰êØUdSÈ)¤.¯¢þ,x=(1;0;


);y=(0;1;0;


);w,kxk=kyk=1:qdx+y=(1;1;0;


);xy=(1;1;0;


);kx+yk=kxyk=21p:p6=2ž,kkx+yk2+kxyk2=221p6=2(kxk2+kyk2)=4;¤±Ø÷v²1o1{K,Klp(p6=2)þ‰êØUdSÈ)¤.(2)C[a;b]þ‰êØUdSÈ)¤.¯¢þ,x(t)=1;y(t)=taba;Kkxk=kyk=1;x(t)+y(t)=1+taba;x(t)y(t)=1taba;kx+yk=2;kxyk=1;¤±kx+yk2+kxyk2=56=2(kxk2+kyk2)=4;C[a;b]þ‰êØUdSÈ)¤.2.y²Ïkxnxk2=(xnx;xnx)=kxnk2(xn;x)(x;xn)+kxk2;(3.0.2)®limn!1kxnk!kxk,d^‡Œ,limn!1(xn;x)!(x;x);limn!1(x;xn)!(x;x);é(3.0.2)ªÒü4,·‚klimn!1kxnxk2=kxk2(x;x)(x;x)+kxk2=0;=limn!1xn=x:
5
3.y²Äky²(
;
)´Enþ˜‡SÈ.(1)é8x2En,(x;x)=nPi;j=1ijxixj.du(ij)´½,(x;x)0,…(x;x)=0žíÑx=0.(2)(1x+2y;z)=nXi;j=1ij(1xi+2yi)zj=nXi;j=1ij1xizj+nXi;j=1ij2xizj=1(x;z)+2(y;z):(3)Ϗ(ij)é¡,ij=ji,(x;y)=nXi;j=1ijxiyj=nXi;j=1ijxiyj=(y;x):nþ,(x;y)´XþSÈ.‡ƒ,e(x;y)XþSÈ,-ij=(ei;ej),8x=nPi=1xiei,k0(x;x)=(nXi=1xiei;nXi=1xjej)=nXi;j=1ijxixj;…x6=0ž,nPi;j=1ijxixj=(x;x)0,¤±(ij)´½.4.y²1x1+2x2+


+nxn=0;Kd(x;x)=8:0;6=;1;=:,éz‡(1n)k=(1x1+2x2+


+nxn;x)=0;¤±x1;x2;


;xn7‚5Ã'.5.y²(1)¿©5.30¦y=x;Kkx+yk=k(1+)xk=(1+)kxk=kxk+kyk:7‡5.kx+yk=kxk+kyk;Kkx+yk2=(kxk+kyk)2=kxk2+kyk2+2kxk
kyk:,˜¡kx+yk2=(x+y;x+y)=kxk2+kyk2+2Re(x;y);
6
1nÙSȘm†Hilbert˜m¤±Re(x;y)=kxk
kyk:lkkykykkxkxk2=2kyk22kykkxkRe(x;y)=0;y=kykkxkx;=y´xê.(2)¿©5.3¦y=x;Kkxyk=k(1)xk=j1jkxk=jkxkkykj:7‡5.kxyk=jkxkkykj;Kkxyk2=(kxkkyk)2=kxk2+kyk22kxk
kykqkxyk2=(xy;xy)=kxk2+kyk22Re(x;y):¤±Re(x;y)=kxk
kyk:lkkykykkxkxk2=2kyk22kykkxkRe(x;y)=0;y=kykkxkx;=y´xê.(3)z6=x;yž,d(1),kxyk=k(xz)+(zy)k=kxzk+kzyk…=30,¦(zy)=(xz).-=1+2(0;1),Kz=x+(1)y:z=x;½z=yž,=0;½=1BŒ.6.y²‡y´±eت:jkXieik2kxk22j=jXij(ei;ej)Xjij2j=jXi6=jij(ei;ej)ja(Xi6=jjijj2)12a(Xjij2jjj2)12=akxk22:7.y²dué82[0;1]kkx+(1)yk=kxk;©O0;12kxk=kyk;…kx+yk=2kxk:ϏH´SȘm,d²1o/{Kkxyk2=2(kxk2+kyk2)kx+yk2=0:
7
x=y:eX´D‰˜mØ´SȘmž,x=yؘ½¤á.~X,3C[0;1]¥,-x(t)1;y(t)=t;Ké82[0;1]kkx+(1)yk=max0t1j+(1)tj=1=kxk:´kxyk=max0t1j1tj=16=0;=x6=y:8.y²-P1n=1xn=M;sk=Pkn=1xn;dksksk0k=kkXn=k0+1xnkkXn=k0+1kxnk(k0k);fskg´Ä,H,K3x2H;÷vx=limk!1sk=1Xn=1xn;…kxk=limk!1kskklimk!1kXn=1kxnk=M:9.y²éfxng;fyng2H;kj1Xn=1(xn;yn)j1Xn=1kxnk
kynk(1Xn=1kxnk2)12
(1Xn=1kynk2)12;¤±(fxng;fyng)=P1n=1(xn;yn)´k¿Â,N´yUù‡SȽÂH´˜‡SȘm.yz‡Hn´Hilbert˜m,·‚y²H´Hilbert˜m.fx(i)g´H¥Ä,Ù¥x(i)=fx(i)1;x(i)2;


;x(i)n;


gK80;3i0;¦i;ji0žkx(i)x(j)k;=1Xn=1kx(i)nx(j)nk22;léz˜‡n;fx(i)ng´Hn¥Ä,limi!1x(i)n=x(0)