泛函分析-孙炯版课后答案-第二章

1ÙD‰˜mSK21.3‚5˜mX¥½Âåld÷v²£ØC5ڃq5,=d(x+z;y+z)=d(x;y);d(x;y)=jjd(x;y):-kxk=d(x;0):y²(X;k
k)´D‰‚5˜m.2.(X;k
k)´D‰‚5˜m,éux;y2X;-d(x;y)=8:0;x=y;kxyk+1;x6=y:y²d´ål,Ø´d‰êpål,=Ø3Xþ‰êk
k1,¦d(x;y)=kxyk1;x;y2X:3.XL«ESx=(x1;x2;


)N,½Âkxk=1Xn=12nmin(1;jxnj):(1)k
k´Ä´Xþ‰ê?(2)kxyk´ÄŒ½ÂXþål?beŒ±,`²kx(n)xk!0(n!1)¿Â.4.3C1[a;b]¥-kxk1=(Zba(jx(t)j2+jx0(t)j2)dt)1=2;8x2C1[a;b]:(1)y²k
k1´C1[a;b]þ‰ê;(2)¯(C1[a;b];k
k1)´Ä?5.3Cn¥½Â‰êkxk=maxijxij;y²§´Banach˜m.6.X´[0;1]þ¤këY¼êx=x(t)8Ü.y²kxk=(Z10jx(t)j2dt)1=2´Xþ‰ê,X3ù«‰êeØ.7.(X;k
k)´D‰˜m,X6=f0g.y²XBanach˜m¿‡^‡´X¥ü ¥¡S=fx2Xkxk=1g.1
2
1ÙD‰˜m8.Ck[a;b]´[a;b]þäkkëYê¼êN.½Âd(f;g)=kXi=0maxx2[a;b]jf(i)(x)g(i)(x)j;f;g2Ck[a;b]:y²(1)(Ck[a;b];d)´ål˜m;(2)e½Âkfk=d(f;0),K(Ck[a;b];k
k)´Banach˜m.9.H´3†‚Rþ²ŒÈ,ꏲŒÈëY¼ê8Ü,éuz‡f2H,½Âkfk=(Z11jf(t)j2dt+Z11jf0(t)j2dt)12:y²H´Banach˜m.10.M´[a;b]þk.¼êN.‚5$Ž½Â†C[a;b]¥ƒÓ.3M¥½Â‰êkxk=supatbjx(t)j.y²M´Banach˜m.11.0p1,ĘmLp[0;1],Ù¥kxk=Z10jx(t)jpdt1;x2Lp[0;1]:y²kxkØ´Lp[0;1]þ‰ê,d(x;y)=kxyk´Lp[0;1]þål.(J«:e0661;K6p61:)12.3l1¥,U‹I½Â‚5$Ž…éx2l1;x=fkg½Âkxk=supnjnj.y²l1´˜‡D‰˜m.13.y²lp(1p1)´Œ©Banach˜m.14.Hp(16p1)L«äk5Ÿkfkp=sup0r1fZ20jf(rei)jpdg1=p1)Û¼êf(z)(jzj1)N.y²kfkp´‰ê¿…(Hp;k
kp)´Banach˜m.15.Hp(0p61)L«[a;b]þN÷vHolder^‡jx(t1)x(t2)j6Mj(t1t2)jp¼ê,‚5$Ž½Â†C[a;b]¥ƒÓ.3Hp¥½Â‰êkxk=jx(a)j+supa6t1t26bjx(t1)x(t2)jjt1t2jp;y²HpBanach˜m.16.x(t)´[a;b]þëY¼ê,-kxkp=(Zbajx(t)jpdt)1p;kxk1=maxatbjx(t)j;y²limp!1kxkp=kxk1:
3
17.y²‚5˜mX¥?ۘxà8E´à8;é?Ûx02X,à8A/£Ä0x0¤8ÜA+x0=fy+x0jy2AgE´à8.18.M‚5˜mX¥f8,y²C0(A)=f1x1+


+nxn2Xjn´?¿g,ê,xk2A;k0…nXk=1k=1g:19.C(0;1]L«(0;1]þëY…k.¼êx(t)N.-kxk=supfjx(t)jj0t1g:y²(1)k
k´C(0;1]˜mþ‰ê;(2)l1†C(0;1]˜‡f˜måÓ.20.Xn‘D‰‚5˜m,E0´Xý4f˜m,y²3x02X,¦kx0k=1,d(x0;E0)=infx2E0kx0xk=1:21.3L2[0;1]þ5½ØӉê:kfk1=Z10jf(t)jdt;kfk2=(Z10jf(t)j2dt)12;kfk3=(Z10(1+t)jf(t)j2dt)12:=´d‰ê?Á`²nd.22.fxng´Banach˜mX¥:.e3(0;1)þšK4~ŒÈ¼êg(t),¦kxnkg(n)(n=1;2;3;


).y²P1n=1xnÂñ.23.(X;k
k)´D‰˜m,Y´Xf˜m,éux2X,·=d(x;Y)=infy2Ykxyk:XJ3y02Y,¦kxy0k=,¡y0´xZ%C.(1)y²XJY´Xk¡‘f˜m.Kéz˜x2X,3Z%C.(2)ÁÞ~`²,YØ´k¡‘˜mž,(1)(Øؤá.(3)ÁÞ~`²,˜„/,Z%C؍˜.(4)y²éuz˜:x2X,x'uf˜mYZ%C:8´à8.24.(Xk;k
kk)´˜D‰˜m,x=fxkg;xk2Xk(k=1;2;


)…÷v^‡P1k=1kxkkpk1;^XL«¤kxN.U‹I½Â‚5$Ž¤‚5˜m,3X¥½Âkxk=(1Xk=1kxkkpk)1=p(p1):y²(X;k
k)´˜‡D‰˜m.25.(X;k
k)D‰˜m,X0´X¥ȗf8.y²éuz˜x2X;3fxngX0,¦x=P1n=1xn,¿…P1n=1kxnk1:
4
1ÙD‰˜m26.X´D‰‚5˜m,M´X4f˜m.y²eXŒ©,KX=MŒ©.27.X´D‰‚5˜m,M´X4f˜m.XJM±9X=M´Banach˜m.y²X´Banach˜m.28.(X1;kxk1);(X2;kxk2)´D‰˜m,3¦È‚5˜mZ=X1X2¥½Âkzk=kx1k1+kx2k2;Ù¥z2Z;z=(x1;x2):f(xn;yn)g´Z¥˜S.y²(1)f(xn;yn)g3Z¥Âñu(x;y),…=fxng3X1¥Âñux,fyng3X2¥Âñuy.(2)f(xn;yn)g´Z¥Cauchy,…=fxng´X1¥Cauchy,fyng´X2¥Cauchy.29.(X1;kxk1);(X2;kxk2)´Banach˜m,3¦È‚5˜mZ=X1X2¥½Âkzk=kx1k1+kx2k2;Ù¥z2Z;z=(x1;x2):y²Z´Banach˜m.30.(X1;kxk1);(X2;kxk2)´D‰˜m,3¦È‚5˜mX1X2¥½Âkzk1=kx1k1+kx2k2;kzk2=maxfkx1k1;kx2k2g:Ù¥z2X1X2;z=(x1;x2):y²k
k1;k
k2´X1X2þd‰ê.
5
1ÙSK{‰1.y²(i)kxk=d(x;0)0(šK5);(ii)kxk=0,d(x;0)=0,x=0(½5);(iii)Šâƒq5d(x;0)=d(x;0)=jjd(x;0);·‚kkxk=d(x;0)=jjd(x;0)=jjkxk(àg);(iv)dålnتkkx+yk=d(x+y;0)d(x+y;y)+d(y;0);2Šâ²£ØC5d(x+z;y+z)=d(x;y);Œd(x+y;y)=d(x;0),kx+ykd(x;0)+d(y;0)=kxk+kyk(nت):¤±(X;k
k)´D‰‚5˜m.2.y²d(x;y)÷všK5!î‚!é¡5´²w.eynت¤á.ex=y,Kd(x;y)=0d(x;z)+d(z;y);Šâ‰ênت,·‚k:d(x;y)=kxyk+1kxzk+kzyk+1:duzØUӞuxÚy,ؔb½x6=z,þª¤:d(x;y)=kxyk+1kxzk+kzyk+1=d(x;z)+kzykd(x;z)+d(z;y):=nت¤á.¤±d(x;y)´ål.Ϗd(x;0)=(kxk+1;x6=0;0;x=0:w,e=2f1;1;0g;x6=0,Kd(x;0)6=jjd(x;0);=Ø÷vàg5.¤±d(x;y)Ø´‰êpål.3.y²(1)k
kØ´Xþ‰ê,Ϗk
kØ÷vàg5.¯¢þ,é=22C;x0=fi;2i;


;ni;


gkkx0k=1Xk=112kmin(1;jxkj)=1Xk=112k=1;
6
1ÙD‰˜mjjkx0k=2(1Xk=112kmin(1;jxkj))=21Xk=112k=2:(2)dk
k½Â´,kxyk÷všK5!½5!é¡5Únت,Kd(x;y)=kxykŒ½ÂXþål,=(X;d)ål˜m.…é?¿fx(n)gX;kx(n)xk!0(n!1)dufx(n)gU‹IÂñx.4.y²(1)k
kw,÷v‰êcn‡^‡(šK5,½5,à5)§2|^MinkowskiتN´yÙ÷vnت.(2)(C1[a;b];k
k1)Ø.ÄC1[a;b]¥¼ê:(xn(t)=r(tc)2+1n2;(atb))1n=1;Ù¥c2(a;b):N´yfxn(t)g1n=1U‰êk
k1´Ä,…::Âñx(t)=jtcj,x(t)62C1[a;b].=3(C1[a;b];k
k1)¥3ØÂñÄfxn(t)g1n=1,l(C1[a;b];k
k1)Ø.5.y²´yk
k´‰ê,e¡y²(Cn;k
k)´.fxkg´Cn¥Cauchy,Ù¥xk=fxk1;xk2;


;xkng.é83K,k;lKžkxkxlk2;=jxkixlij2;i=1;2;


;n:ùL²éuz˜‡i(i=1;2;


;n),fxkig1k=1´C¥Cauchy.dC53xi2C¦xki!xi;(k!1):-x=fx1;x2;


;xng,w,x2Cn:dujxkixlij2;(i=1;2;


;n),ü-l!1jxkixij2;(i=1;2;


;n);kxkxk=max1injxkixij:KfxkgÂñ,=Cn´.
7
6.y²(1)k
kw,÷v‰êcn‡^‡(šK5,½5,à5),2|^MinkowskiتN´yÙ÷vnت.(2)e¡·‚E˜‡ù‡˜m¥Cauchy,§3ù‡˜m¥ØÂñ.ÄëY¼êfxn(t)g(n2):xn(t)=8:0;0t121n;1;12t1;†‚ë;Ù§.´,fxng´X¥Cauchy.eX,K3X¥ëY¼êy(t),¦d(xn;y)!0;(n!1):Äx(t)=8:0;0t12;1;12t1;12;t=12:dnتZ10jx(t)y(t)jdt=0;=x(t)A??uy(t).Ϗx(t)Úy(t)3[0;12)[(12;1]þëY,ü‡ëY¼êA??ƒ,Ò´::ƒ.u´3[0;12)þy(t)=x(t)=0;3(12;1]þy(t)=x(t)=1;Ky(t)3t=12:ØëY(†m4Øƒ),†y(t)ëYgñ.7.y²7‡5:fxngSÄ,ϏX,¤±3x2X¦xn!x;Kd‰êëY5,·‚kkxk=limn!1kxnk=1;x2S,=S.¿©5:S,?X¥Äfxng,Ϗjkxnkkxmkjkxnxmk!0(n;m!1);¤±fkxnkgÂñê.(1)elimn!1kxnk=0,Klimn!1xn=0;(2)elimn!1kxnk=a0,Kn¿©Œžkxnka20,-x0n=