泛函分析 孙炯版答案 第三章

1nÙSȘm†Hilbert˜mSK31.Þі�ü‡‚5D‰˜mX,¦�3Xþ�‰êØUdSÈ)¤.2.�{xn}SȘmH¥:�.‖xn‖→‖x‖,…(xn,y)→(x,y)(y∈H)(n→∞),y²:xn→x(n→∞).3.�En´n‘‚5˜m,{e1,e2,···,en}´En�˜‡Ä,(αij)(i,j=1,2,···,n)´�½Ý,éEn¥��ƒx=∑ni=1xiei9y=∑ni=1yiei,½Â(x,y)=n∑i,j=1αijxiyj,(3.0.1)K(·,·)´Enþ�˜‡SÈ.‡ƒ,�(·,·)´Enþ�˜‡SÈ,K73�½Ý(αij)¦�(3.0.1)¤á.4.�H´SȘm,x1,x2,···,xn´H¥��ƒ,§‚÷v^‡(xµ,xγ)=0,�µ6=γ,1,�µ=γ.y²{x1,x2,···,xn}´‚5Ã'�.5.�x,y´ESȘmX¥�ü‡š"�ƒ,K(1)‖x+y‖=‖x‖+‖y‖�…=�y´x���ê;(2)‖x−y‖=|‖x‖−‖y‖|�…=�y´x���ê;(3)‰½z∈X,‖x−y‖=‖x−z‖+‖z−y‖�…=�3α∈[0,1],¦�z=αx+(1−α)y.6.�ei∈X,‖ei‖=1(i∈N),a2=∑i6=j∣∣(ei,ej)∣∣2<∞,x={λi}∈l2,K(1−a)‖x‖22≤‖∑λiei‖2≤(1+a)‖x‖22.7.�XSȘm,x,y∈X,b½‖λx+(1−λ)y‖=‖x‖,∀λ(06λ61).y²x=y.eX´D‰˜m�Ø´SȘmž,œ¹qXÛ?8.�H´Hilbert˜m,{xn}⊂H,÷v∑∞n=1‖xn‖<∞.y²∑∞n=1xn3H¥Âñ.9.�{Hn}(n=1,2,···)˜�SȘm.-H={{xn}|xn∈Hn,∞∑n=1‖xn‖2<∞}.1·2·1nÙSȘm†Hilbert˜méu{xn},{yn}∈H.½Âα{xn}+β{yn}={αxn+βyn}(α,β∈K),({xn},{yn})=∞∑n=1(xn,yn).y²H´SȘm,¿…�z˜‡HnÑ´Hilbert˜mž,H´Hilbert˜m.10.éuSȘmH,eã^‡�dµ(1)x⊥y;(2)‖x+αy‖>‖x‖,∀α∈C;(3)‖x+αy‖=‖x−αy‖,∀α∈C.11.eSȘmX´¢�,K‖x+y‖2=‖x‖2+‖y‖2%¹Xx⊥y,�eX´E˜mž,x⊥y™7¤á.Þ~`²ƒ.12.�M={x|x={xn}∈l2,x2n=0,n=1,2,···},y²M´l2�4f˜m,…¦ÑM⊥.13.�X=Rk,A={a},Ù¥a=(a1,···,ak),y²A⊥={(x1,···,xk)∈Rk∣∣k∑j=1ajxj=0}.14.�X´SȘm,A⊂X.y²A⊥=A¯⊥.15.�XÚY´Hilbert˜mH�‚5f˜m,-X+Y={x+y|x∈X,y∈Y}.y²(X+Y)⊥=X⊥∩Y⊥.16.3L2[a,b]¥,-S={e2piinx}∞n=−∞.(1)e|b−a|≤1.¦yS⊥={0};(2)e∣∣b−a∣∣>1.¦yS⊥6={0}.17.�M,N´SȘmH�f˜m,M⊥N,L=M⊕N,y²L´4f˜m�¿©7‡^‡´M,Nþ4f˜m(¿©5Ü©b½H��).18.3C[−1,1]=X¥.-(1)M1={f∈X|f(x)=0,∀x<0};(2)M2={f∈X|f(0)=0}.OŽM1,M23X¥'uSÈ(f,g)=∫1−1f(x)g(x)dx���Ö.19.eH´SȘm,M,N⊂H,K(1)eM⊥N,KM⊂N⊥,N⊂M⊥;(2)M⊥=(M)⊥.20.�(X,‖·‖)´D‰˜m.(1)y²D‰˜m(X,‖·‖)´î‚à�,�…=�,é?¿x,y∈X,x6=0,y6=0,·3·‖x+y‖=‖x‖+‖y‖7kx=αy(α>0).(2)y²3î‚àD‰˜m¥,éuz‡x∈X,x'u?¿4f˜mY�Z%C´˜�.21.D‰‚5˜mX�¡Š´˜—à�,eX¥�?Û÷v‖xn‖=‖yn‖=1,‖xn+yn‖→2�S�{xn},{yn}k‖xn−yn‖→0.(1)y²?ÛSȘmÑ´˜—à�.(2)C[a,b]Ø´˜—à�.(3)L1[a,b]Ø´˜—à�.22.�{x1,x2,x3}´SȘmX¥�‚5Ã'8,b½{x1,x2}÷v(xi,xj)=δij,16i,j62.½Âf:C2→RXe:f(α1,α2)=‖α1x1+α2x2−x3‖.y²�αi=(x3,xi),i=1,2ž,f���Š.23.�{eα}(α∈I)´SȘmH¥�IO��X.y²éuz‡x∈H,x'uù‡IO��X�FourierXê{(x,eα)|α∈I}¥õkŒê‡Ø".24.M´H�4‚5f˜m,{en}†{e′n}©O´M†M⊥�IO��Ä,y²{en}∪{e′n}�¤H�IO��Ä.25.�HHilbert˜m,eE⊂H´‚5f˜m¿…éu?¿�x∈H,x3Eþ�ÝK3,KE´4�.26.�A={ek}´SȘmX¥�IO��X.y²é∀x,y∈X,k∞∑k=1∣∣(x,ek)(y,ek)∣∣≤‖x‖‖y‖.27.�HHilbert˜m,{ek},{e′k}´H¥�ü‡IO��X,¿…∞∑k=1‖ek−e′k‖2<1.y²XJ{ek},{e′k}¥ƒ˜´���,K,˜‡´���.28.Þ~`²SȘm¥���IO��Xؘ½´���.29.¡Hn(t)=(−1)net2dndtne−t2Hermiteõ‘ª,-en(t)=(2nn!√pi)−1/2e−t22Hn(t),(n=1,2,3,···),y²{en}|¤L2(−∞,+∞)¥�˜‡���IO��X.30.-Ln(t).X�(Laguerre)¼êetdndtn(tne−t),y²{1n!e−t/2Ln(t)}(n=1,2,···)|¤L2(0,∞)¥�˜‡���IO��X.31.y²3Œ©�SȘm¥,?˜IO��Xõ˜Œê8.·4·1nÙSȘm†Hilbert˜m1nÙSK{‰1.)(1)lp(p6=2)þ�‰êØUdSÈ)¤.¯¢þ,�x=(1,0,···),y=(0,1,0,···),w,‖x‖=‖y‖=1.qdx+y=(1,1,0,···),x−y=(1,−1,0,···),�‖x+y‖=‖x−y‖=21p.�p6=2ž,k‖x+y‖2+‖x−y‖2=2×21p6=2(‖x‖2+‖y‖2)=4,¤±Ø÷v²1o>1{K,Klp(p6=2)þ�‰êØUdSÈ)¤.(2)C[a,b]þ�‰êØUdSÈ)¤.¯¢þ,�x(t)=1,y(t)=t−ab−a,K‖x‖=‖y‖=1,x(t)+y(t)=1+t−ab−a,x(t)−y(t)=1−t−ab−a,‖x+y‖=2,‖x−y‖=1,¤±‖x+y‖2+‖x−y‖2=56=2(‖x‖2+‖y‖2)=4,�C[a,b]þ�‰êØUdSÈ)¤.2.y²Ï‖xn−x‖2=(xn−x,xn−x)=‖xn‖2−(xn,x)−(x,xn)+‖x‖2,(3.0.2)®limn→∞‖xn‖→‖x‖,d^‡Œ,limn→∞(xn,x)→(x,x),limn→∞(x,xn)→(x,x),é(3.0.2)ª�Òü>�4,·‚klimn→∞‖xn−x‖2=‖x‖2−(x,x)−(x,x)+‖x‖2=0,=limn→∞xn=x.·5·3.y²Äky²(·,·)´Enþ�˜‡SÈ.(1)é∀x∈En,(x,x)=n∑i,j=1αijxixj.du(αij)´�½,�(x,x)≥0,…�(x,x)=0žíÑx=0.(2)(β1x+β2y,z)=n∑i,j=1αij(β1xi+β2yi)zj=n∑i,j=1αijβ1xizj+n∑i,j=1αijβ2xizj=β1(x,z)+β2(y,z).(3)Ϗ(αij)é¡�,αij=αji,�(x,y)=n∑i,j=1αijxiyj=n∑i,j=1αijxiyj=(y,x).nþ,(x,y)´Xþ�SÈ.‡ƒ,e(x,y)XþSÈ,-αij=(ei,ej),∀x=n∑i=1xiei,k0≤(x,x)=(n∑i=1xiei,n∑i=1xjej)=n∑i,j=1αijxixj,…�x6=0ž,n∑i,j=1αijxixj=(x,x)>0,¤±(αij)´�½.4.y²�α1x1+α2x2+···+αnxn=0,Kd(xµ,xγ)=0,�µ6=γ,1,�µ=γ.,éz‡γ(1≤γ≤n)kαγ=(α1x1+α2x2+···+αnxn,xγ)=0,¤±x1,x2,···,xn7‚5Ã'.5.y²(1)¿©5.�3λ>0¦y=λx,K‖x+y‖=‖(1+λ)x‖=(1+λ)‖x‖=‖x‖+‖y‖.7‡5.�‖x+y‖=‖x‖+‖y‖,K‖x+y‖2=(‖x‖+‖y‖)2=‖x‖2+‖y‖2+2‖x‖·‖y‖.,˜¡‖x+y‖2=(x+y,x+y)=‖x‖2+‖y‖2+2Re(x,y),·6·1nÙSȘm†Hilbert˜m¤±Re(x,y)=‖x‖·‖y‖.l k‖y−‖y‖‖x‖x‖2=2‖y‖2−2‖y‖‖x‖Re(x,y)=0,�y=‖y‖‖x‖x,=y´x���ê.(2)¿©5.�3λ¦y=λx,K‖x−y‖=‖(1−λ)x‖=|1−λ|‖x‖=|‖x‖−‖y‖|.7‡5.�‖x−y‖=|‖x‖−‖y‖|,K‖x−y‖2=(‖x‖−‖y‖)2=‖x‖2+‖y‖2−2‖x‖·‖y‖q‖x−y‖2=(x−y,x−y)=‖x‖2+‖y‖2−2Re(x,y).¤±Re(x,y)=‖x‖·‖y‖.l k‖y−‖y‖‖x‖x‖2=2‖y‖2−2‖y‖‖x‖Re(x,y)=0,�y=‖y‖‖x‖x,=y´x���ê.(3)�z6=x,yž,d(1),‖x−y‖=‖(x−z)+(z−y)‖=‖x−z‖+‖z−y‖�…=�3λ>0,¦�(z−y)=λ(x−z).-α=λ1+λ∈(0,1),Kz=αx+(1−α)y.�z=x,½z=yž,�α=0,½α=1BŒ.6.y²‡y�´±eØ�ª:|‖∑λiei‖2−‖x‖22|=|∑λiλj(ei,ej)−∑|λi|2|=|∑i6=jλiλj(ei,ej)|≤a(∑i6=j|λiλj|2)12≤a(∑|λi|2|λj|2)12=a‖x‖22.7.y²dué∀λ∈[0,1]k‖λx+(1−λ)y‖=‖x‖,�λ©O�0,12�‖x‖=‖y‖,…‖x+y‖=2‖x‖.ϏH´SȘm,d²1o>/{K�‖x−y‖2=2(‖x‖2+‖y‖2)−‖x+y‖2=0.·7·�x=y.eX´D‰˜m�Ø´SȘmž,x=yؘ½¤á.~X,3C[0,1]¥,-x(t)≡1,y(t)=t,Ké∀λ∈[0,1]k‖λx+(1−λ)y‖=max0≤t≤1|λ+(1−λ)t|=1=‖x‖.�´‖x−y‖=max0≤t≤1|1−t|=16=0,=x6=y.8.y²-∑∞n=1xn=M,sk=∑kn=1xn,d‖sk−sk′‖=‖k∑n=k′+1xn‖≤k∑n=k′+1‖xn‖(k′<k),{sk}´Ä��, H��,K3x∈H,÷vx=limk→∞sk=∞∑n=1xn,…‖x‖=limk→∞‖sk‖≤limk→∞k∑n=1‖xn‖=M.9.y²é{xn},{yn}∈H,k|∞∑n=1(xn,yn)|≤∞∑n=1‖xn‖·‖yn‖≤(∞∑n=1‖xn‖2)12·(∞∑n=1‖yn‖2)12,¤±({xn},{yn})=∑∞n=1(xn,yn)´k¿Â�,N´�yUù‡SÈ�½ÂH´˜‡SȘm.y�z‡Hn´Hilbert˜m,·‚y²H´Hilbert˜m.�{x(i)}´H¥�Ä��,Ù¥x(i)={x(i)1,x(i)2,···,x(i)n,···}K∀ε>0,3i0,¦i,j≥i0ž‖x(i)−x(j)‖<ε,=∞∑n=1‖x(i)n−x(j)n‖2<ε2,l éz˜‡n,{x(i)n}´Hn¥�Ä��,�limi→∞x(i)n=x(0)n(n=1,2,3,···),Px={x(0)1,x(0)2,···,x(0)n,···},y²x∈H,…‖x(i)−x‖→0(i→∞).Ϗé?¿�g,êk,�i,j≥i0žk∑n=1‖x(i)n−x(j)n‖2<ε2.·8·1nÙSȘm†Hilbert˜m-j→∞,K�i≥i0žk∑n=1‖x(i)n−x(0)n‖2<ε2,2-k→∞,�∞∑n=1‖x(i)n−x(0)n‖2<ε2,(i≥i0).u´x=x(i)−(x(i)−x)∈H…limi→∞‖x(i)−x‖=0,�HHilbert˜m.10.y²(1)⇒(2),(1)⇒(3)w,¤á.(2)⇒(1).e‖x+αy‖≥‖x‖,…y6=0,K�α=−(x,y)‖y‖2,Œ�0≤(x+αy,x+αy)−‖x‖2=α(x,y)+α(y,x)+|α|2‖y‖2=−|(x,y)|2‖y‖−2≤0�7kx⊥y.(3)⇒(1).e‖x+αy‖≥‖x−αy‖,K(x+αy,x+αy)=(x−αy,x−αy),Ðm�Reα(x,y)=0©O�α=1,α=i,�(x,y)�¢ÜJÜþ0,=(x,y)=0.11.y²3¢SȘmS,‖x+y‖2=(x+y,x+y)=‖x‖2+2(x,y)+‖y‖2,�‖x+y‖2=‖x‖2+‖y‖2�…=�(x,y)=0,=x⊥y.e´ESȘm,x⊥yؘ½¤á.~X3C¥�x=1+i,y=1−i,K‖x+y‖2=‖x‖2+‖y‖2,�(x,y)6=0.12.y²w,M´l2�‚5f˜m.e¡yM=M.´M⊆M,IyM⊆M.é∀x={xn}∈M,∃{xk}⊆M¦�xk→x.=‖xk−x‖={∞∑n=1|xkn−xn|2}12→0,k→∞.l éu∀n,xkn→xn(k→∞).dxk∈M�,én=2m(m=1,2,···)xk2m=0,Ïdx2m=0.ùL²x∈M.�M⊆M.KM´l2�4f˜m.-N={x|x={xn}∈l2,x2n−1=0,n=1,2,···}.e¡y²M⊥=N.éu∀x={xn}∈N,∀y={yn}∈M,k(x,y)=∞∑n=1xnyn=0,·9·l x∈M⊥,=N⊆M⊥.éu∀x∈M⊥,�{yk}⊆M,÷vykn={1,n=2k−1;0,n6=2k−1,K(x,yk)=∞∑n=1xnykn=x2k−1=0,(k=1,2,···).Ïdx∈N,=M⊥⊆N.�M⊥={x|x={xn}∈l2,x2n−1=0,n=1,2,···}.13.y²ÏA⊥={x=(x1,···,xk)∈Rk|(x,a)=0},…(x,a)=∑kj=1ajxj,¤±�A⊥={(x1,···,xk)∈Rk∣∣k∑j=1ajxj=0}.14.y²ÏA⊂A¯,¤±A⊥⊃A¯⊥.‡ƒ,?�x∈A⊥.é?¿�y∈A¯,3{yn}⊂A,¦�yn→y(n→∞).Ïd(x,y)=limn→∞(x,yn)=0,l x∈(A¯)⊥.�,A⊥⊂(A¯)⊥.nþ=�A⊥=(A¯)⊥.15.y²?�z∈(X+Y)⊥,é∀x+y∈X+Y,k(z,x+y)=0.©O�y=0,x=0,�(z,x)=0∀x∈X,(z,y)=0∀y∈Y,�z∈X⊥∩Y⊥,=(X+Y)⊥⊂(X⊥∩Y⊥).,˜¡,?�z∈X⊥∩Y⊥,Ké∀x∈X,y∈Y,k(z,x)=0,(z,y)=0.�é∀x+y∈X+Y,k(z,x+y)=0,=z∈(X+Y)⊥.¤±(X⊥∩Y⊥)⊂(X+Y)⊥.nþ¤ã,(X+Y)⊥=(X⊥∩Y⊥).16.y²éu∀n,e2piinx�±Ï´1.(1)e|b−a|=1,{e2piinx}∞n=−∞´L2[a,b]þ�˜|��Ä,�S⊥={0}.e|b−a|<1,é∀u∈S⊥⊂L2[a,b],-u˜={u,x∈[a,b]0,x∈[b,a+1]Kd∫a+1au˜e2piinxdx=0·10·1nÙSȘm†Hilbert˜mu˜=0(x∈[a,a+1]),=u=0(x∈[a,b]).�S⊥={0}.(2)e|b−a|>1,ùž{e2piinx}∞n=−∞´L2[b−1,b]þ�˜|��Ä.Ïd,L2[b−1,b]þ�¼êŒ±d§Fourier�Xêû½.Ké∀u∈L2[a,b−1],u6=0,Œò§*¿L2[a,b]þ�¼êv(x)∈S⊥, v(x)6=0.¯¢þ,-v={u(x),x∈[a,b−1]u˜(x),x∈[b−1,b]Ù¥(b−1,b]þ�¼êu˜(x)�Fp“XêÏLu(x)3[a,b−1]þ�Š5OŽ,=u˜n=∫bb−1u˜e2piinxdx=−∫b−1aue2piinxdx,u´u˜=∞∑n=−∞u˜ne2piinx∈L2[b−1,b],¿…∫bave2piinxdx=∫b−1aue2piinxdx+∫bb−1u˜e2piinxdx=0.=v(x)∈S⊥.17.y²7‡5.�L4f˜m,{xn}⊂M,xn→x(n→∞),Kx∈L…阃y∈Nk(x,y)=limn→∞(xn,y)=0,¤±x∈N⊥.ϏL=M⊕N…x∈L,x∈N⊥,�x∈M.¤±M4f˜m.ÓnŒyN4f˜m.¿©5.�{x(n)}⊂L,x(n)→x∈H(n→∞),-x(n)=x(n)1+x(n)2,x(n)1∈M,x(n)2∈N,éxŠ��©):x=x1+x2(x1∈M,x2∈M⊥).Ϗ‖x(n)i−xi‖≤‖x(n)−x‖→0(i=1,2)dM,N4Œx1∈M,x2∈N,�x∈L=L´4f˜m.18.)(1)M⊥1={g∈X|∫1−1fgdx=0,∀f∈M1}={g∈X|∫0−1fgdx+∫10fgdx=0,∀f∈M1}={g∈X|∫10fgdx=0,∀f∈M1}={g∈X|g(x)=0,1≥x>0}·11·(2)M⊥2={g∈X|∫1−1fgdx=0,∀f∈M2}=é?¿�f(x)∈X…f(0)=0,Ñk∫1−1fgdx=0.?�f(x)∈X,…f(x)=0,−1≤x≤0,d(1)Œíg(x)=0,0≤x≤1.Ón,?�f(x)∈X,…f(x)=0,0≤x≤1,naqu(1),²OŽŒ�g(x)=0,−1≤x≤0.nþŒ�M⊥2={g(x)≡0,x∈[−1,1]}.19.y²(1)ϏM⊥={x:(x,y)=0,∀y∈M},N⊥={x:(x,y)=0,∀y∈N},N⊥M,¤±N⊂M⊥,M⊂N⊥.(2)ϏM⊂M,¤±M⊥⊂M⊥.‡ƒ,é∀z∈M3{xn}⊂M¦�xn→z(n→∞).l é∀y∈M⊥,k(y,z)=(y,limn→∞xn)=limn→∞(y,xn)=0.�y∈(M)⊥,=M⊥⊂(M)⊥.KM⊥=(M)⊥.20.y²(1)é∀x,y∈X,x6=y¿…‖x‖=‖y‖=1,k‖αx+βy‖≤1,∀α,β>0,α+β=1.e‖αx+βy‖=1,K3α˜>0¦�αx=α˜βy,u´1=‖x‖=α˜βα‖y‖=α˜βα,l �x=y,gñ.�D‰˜m(X,‖·‖)´î‚à�.‡L5,�x,y´X¥š"�¦�‖x+y‖=‖x‖+‖y‖.2�a=‖x‖,b=‖y‖,x1=xa,y1=yb,Ka>0,b>0,‖x1‖=‖y1‖=1,…‖aa+bx1+ba+by1‖=1,duD‰˜m(X,‖·‖)´î‚à�,�x1=y1,=x=αy,Ù¥α=ab.(2)b�x0,y0Ñ´x'uY�Z%C,Kd∆===infy∈Y‖x−y‖=‖x−x0‖=‖x−y0‖.·12·1nÙSȘm†Hilbert˜mdx0,y0∈Yx0+y02∈Y,l ‖x−x0+y02‖≥d.,˜¡,duD‰˜m(X,‖·‖)´î‚à�,�‖x−x0+y02‖=d‖x−x02d+x−y02d‖<d,gñ,Kx0=y0.21.y²(1)�{xn}⊂X,{yn}⊂X,÷v‖xn‖=‖yn‖=1,‖xn+yn‖→2.d²1o>/{KŒ‖xn−yn‖2=2(‖xn‖2+‖yn‖2)−‖xn+yn‖2=4−‖xn+yn‖2,�limn→∞‖xn−yn‖=limn→∞(4−‖xn+yn‖2)12=0.KSȘm´˜—à�.(2)3C[a,b]¥,�xn(t)=1,yn(t)=t−ab−a,K‖xn‖=‖yn‖=‖xn+yn‖2=‖xn−yn‖=1.w,,‖xn+yn‖→2(n→∞),�´‖xn−yn‖=190(n→∞).�C[a,b]Ø´˜—à�.(3)3L[a,b]¥,�xn(t)=1b−a,yn(t)=2(t−a)(b−a)2,K‖xn‖=‖yn‖=‖xn+yn‖2=1,‖xn−yn‖=12.w,,‖xn+yn‖→2(n→∞),�´‖xn−yn‖=1290(n→∞).�L[a,b]Ø´˜—à�.22.y²f(α1,α2)=‖α1x1+α2x2−x3‖=‖(x3,x1)x1+(x3,x2)x2−x3+(α1−(x3,x1))x1+(α2−(x3,x2))x2‖Ϗ((x3,x1)x1+(x3,x2)x2−x3,(α1−(x3,x1))x1+(α2−(x3,x2))x2)=0¤±((x3,x1)x1+(x3,x2)x2−x3)⊥((α1−(x3,x1))x1+(α2−(x3,x2))x2))¤±f(α1,α2)=‖α1x1+α2x2−x3‖+‖(α1−(x3,x1))x1+(α2−(x3,x2))x2‖·13·qϏx1⊥x2�f(α1,α2)=‖α1x1+α2x2−x3‖+|α1−(x3,x1)|‖x1‖+|α2−(x3,x2)|‖x2‖¤±,�α1=(x3,x1),α2=(x3,x2)ž,f��Š.23.y²∀x∈X,ؔ�x6=0,-F={eα|(x,eα)6=0,α∈I}={eα||(x,eα)|>0,α∈I}?˜Ú·‚kF=∪∞n=1{eα||(x,eα)|>1n,α∈I}PFn={eα||(x,eα)|>1n,α∈I},w,Fn�‡êج‡LN=n2[‖x‖2+1]([·]L«˜‡ê��êÜ©),=Fn�‡êk,beØ,,kei1,ei2,···,eiN÷v|(x,eik)|>1n,k=1,2,···,N,KN∑k=1|(x,eik)|2>N·1n2≥‖x‖2,†�l�Ø�ªN∑k=1|(x,eik)|2≤‖x‖2,gñ.duŒê‡–õŒê8Ü�¿8´Œê�,F=∪∞n=1Fn´Œê8.24.y²é∀x∈H,d��©)½n,x=y+z,y∈M,z∈M⊥.ìK¿,y=∞∑n=1(y,en),z=∞∑n=1(z,e′n).�é∀x∈Hkx=y+z=∞∑n=1(y,en)+∞∑n=1(z,e′n).ddŒ„,{en}∪{e′n}�¤H�IO��Ä.25.y²�{xn}⊂E,…limn→∞xn=x.Ϗx3Eþ�ÝK3,�3x0∈E,x1⊥E¦�x=x0+x1.Ϗxn∈E,¤±(xn,x1)=0,l0=limn→∞(xn,x1)=(x,x1)=(x0+x1,x1)=(x1,x1)�x1=0,=x=x0∈M,KE´4.·14·1nÙSȘm†Hilbert˜m26.y²duA={ek}´SȘmX¥�IO��X.�(ÜHo¨lderØ�ªÚBesselØ�ª�∞∑n=1|(x,en)(y,en)|≤[∞∑n=1|(x,en)|2]12[∞∑n=1|(y,en)|2]12≤‖x‖‖y‖.27.y²�{ek}��,·‚y²{e′k}��.be{e′k}Ø��,K73š"�ƒx0∈H,¦�x0⊥e′k,k=1,2,···.qd{ek}´���,�‖x0‖2=∞∑k=1|(x0,ek)|2.u´‖x0‖2=∞∑k=1|(x0,ek)|2=∞∑k=1|(x0,ek)−(x0,e′k)|2=∞∑k=1|(x0,ek−e′