泛函分析刘炳初习题答案后清晰版.pdf

4SKY() SK 1! y 1)eyvlIv?n w,d(x,y) 0,d(x,y) = 0 sup|xy| = 0,sup|x 0y0| = 0 3Dx = y. w,d(x,y) = d(y,x). d(x,y) = sup|xy|+sup|x 0 y 0| sup|xz|+sup|z y|+sup|x0 z 0|+ sup|z 0 y 0| = d(x,z) + d(z,y). 2)D:?xnUlk?=?xnx 0 n. 3)?xn D,vxn C0,1x 0 n C0,1Cauchy ?.K 0,N,m,n N,t 0,1,|xn(t) xm(t)| N,|xn(t) x(t)| + |x 0 n(t) x 0(t)| 2. =3Dxn x,l?D?. 4! ?n:3S?)?U3. ?. 1)d(x,y) 0,d(x,y) = 0 max |t|1 |x y| = max |t|=1 |x y| = 0 x = y 2)w,d(x,y) = d(y,x). 3)d(x,y) = max |t|=1 |x y| max |t|=1 |x z| + max |t|=1 |z y| = d(x,z) + d(z,y). 5!“ :?X = 0,2.8,5.6,UR?llm.w, 1 O(0,4) = 0,2.8 (2)T : B B 1)B X?B 4,X?,B?. 2x B,eyTx B,=yd(Tx,x0) r d(Tx,x0) d(Tx,Tx0) + d(Tx0,x0) d(x,x0) + (1 )r r + (1 )r = r. 19!?d = diam(N),f(t,s)3(t0,s0)?d?Y. K, N1, kxk x0k = sup n |(k) n 0 n| 2 du |0 n| | k n| + | k n 0 n| N1) qxk c0,k n 0(n ),?3N N1,?n Nk|(k) n| 2, l?|0 n| N,=x0 c0ddc0l?4fm. 3).dulBanachm,?c0l?4fm,l?c0Banachm,eyc0 ?. ?Mkn?N,=x = (1,2, ,n,) M n?kn ,3Nx,?n Nx,n= 0.w,M S n=1 Qn,M. 0,y = (1,2, ,n,) c0 du 0,?3N,?n Nn M d(.mE1 0,l? R E |x(t)|pdt R E1 |x(t)|pdt (M )p mE1 -p K lim p kxkp M 14 d?5, lim p kxkp M,l? lim p kxkp= M = kxk 13!?(X1,kk1),(X2,kk2)Dm,35mX1X2kzk1= kx1k1+ kx2k2,kzk2= max(kx1k1,kx2k2),z X1 X2,z = (x1,x2)y kzk1,kzk2,X1 X2?d. y:w,kzk2 kzk1 2kzk2,l?d. 14!?Xma,bkYU5$? 5mux X,kxk = sup atb |x(t)|;kxk1= R b a |x(t)|dt.y:k kk k1 X?d?. y:b?d,w,dkk?m?,?kk1?m ?,g. 15!?Banachm(X,kk)kSchauderen,ML? P k=1 kek 3X?k?N,U5$?5 m,uzx = k M,kxk1= sup n k P k=1 kekk,y(M,kk)Banach m y:k?Cauchy?. 16!?(X,kk)Dm,Y X?fm,ux X- = d(x,Y ) = inf yY kx yk.XJ3y0 Y ,?kx y0k = y0x?Z%C. 1)y:XJY X?kfm,Kzx X,3Z%C. 2),?Y km,1)?(. 3),/,Z%C. 4)yuz:x X,xufmY ?Z%C:88. y: 1)de(., 0,yn Y,? kx ynk ?e(.,?d(x0,M) 1,l?d(x0,M) = 1,=3M?:?T:x0? Z%C. 3):3R2x = (x1,x2) R2, kxk = max(|x1|,|x2|) ?x0= (0,1),e1= (1,0),a R,K kx0 ae1k = k(a,1)k = max|a|,1 l?min aR kx0 ae1k = 1?Z%C?ae1|a|1. 4)?MxufmY ?Z%C:8K y1,y2 M, 0,1,kx y1k = kx y2k = d(x,Y ) l? kx (y1+ (1 )y2)k = k(x y1) + (1 )(x y2)k kx y1k + (1 )kx y2k = d(x,Y ) qw, k(x y1) + (1 )(x y2)k d(x,Y ) l? k(x y1) + (1 )(x y2)k = d(x,y) = y1+ (1 )y2 M M8. 17!?(X,kk)Dm,XJ?x,y X,x 6= ykxk = kyk = 1 16 7kkx + yk 0) 2)y3Dm,uzx X,xu?fmY ?Z% C?. y 1)75.ex,yvkx + yk = kxk + kyk.X?,e x kxk 6= y kyk, 7kk x kxk + y kykk 0, ?kxyk ,kxk = kyk = 1,7kkx yk 2 , (X,kk)?. y 1)Dm7?. 2)Ca,b?. 17 3)L1a,b?. y:?X?Dm,x,y X,x 6= ykxk = kyk = 1K7 30 0,?kx yk 0. (e, 0 kkx yk 073 0,?kx yk 2 1 2,n = 1,2, . b?X.x 0 n n=1m?fmX.KAkx 0 0 X 0, |x 0 0(xn)| = 0,n = 1,2, . kx 0 0k = 1l? kx 0 0 x 0 nk |x 0 0(xn) x 0 n(xn)| = |x 0 n(xn)| 1 2,n = 1,2, . x 0 n n=1X 0 ?f8g.y 6! Tn: Lp C y(t) R b a xn(t)y(t)dt. kTnk = kxnkp 1. ? 21 yn= xn f(xn) x1 f(x1) K f(yn) = 0 ? yn x1 f(x1),f( x1 f(x1) = 1. N(f)48g.l?fk.f. 14! F1,F2 Xf?,3x / M,F1(x) 6= F2(x).o,0 1,F= F1+ (1 )F2f?,16= 2,F16= F2,l?f? ?N?7?uY. 15! 75:df?Y5w,. 5: 1x0 G,dHahnBanachn,g spanx0,G,?g|G= 0g(x0) = 1.g. 16! 3spanx1, ,xnf v f( n P k=1 tkxk) = n P k=1 tkak. ?fg,v1)2).2?F Xf?=. 17!?xnX?f8.fnv fn(xn) = kxnkkfnk = 1. ? = fn: n 1. ,sup f |f(x)| kxk. , x X,xi x,|fi(x) fi(xi)| 0 |f(xi)| kxik 1 n. n,kxk = sup f |f(x)|. 18!d19K,x0 spanx1, ,xn,?73xn?k5|r ux0 19!1x0/ M,dHahn-Banachn,g X,g(x0) 0 g(M) = 0. 22 xnfux0g. 20!?XDm,x0,xn X(n = 1,2,).yXJxn w x0(n ),kxnk kx0k(n )Kxn x0(n ). y:”?kx0k 6= 0,xn6= 0,y. ,?kxnk = kx0k = 1, xnUu0 o? 0, ?kxn x0k 0,dm?5 0,?kxn+ x0k 2 . duxn w x0,?f X,kfk = 1,kf(xn) f(x0),l?k f(xn+ x0) 2f(x0). du |f(xn+ x0)| |f|kxn+ x0k 2 9 kx0k = sup kfk=1 lim n 1 2|f(xn + x0)| 2 2 kx0k 0,s.t.|Sn| C(n 1). 25 kyn= P k=1 nkSkm?.d(3)? P k=1k.,l ?.Cauchy?n, 0,K,s.t. K2 K1 K K2 X k=K1 |nk| K1 K, | K2 X k=K1 nkSk| K2 X k=K1 |nk|Sk| C K2 X k=K1 |nk| 0,3K ? X k=K+1 |nk| 4C . TK.d(1), N = N(,K),s.t.n N |nk| 4CK (k = 1,2, ,K). ?,?n N, | X k=1 nk(Sk S)| K X k=1 |nk|(Sk S)| + X k=K+1 |nk|(Sk S)| 4CK K X k=1 |(Sk S)| + 2C X k=K+1 |nk| i0,kkx(i) x(j)k 2 1 2 = 1, =eG?ALO(xn, 1 2)?.=eG. 7!?e, ISmH?IO?X.yuzx H,x uIO?X?FourierX(x,e) : Ik“. y:?x HdSchwarz?,z I,|(x,x)| kxk.5 S k=1 1 k+1kxk, 1 kkxk = (0,kxk. u,ek(x,xn) 6= 0,K?km 1 k+1kxk, 1 kkxk |(x,x)|. Bessel?g.y 8!?HHilbertm,x0,xn H(n = 1,2,).?n ,xn w x0,kxnk kx0k,(n ). y:d kxn xk2= (x xn,x xn) = (x,x) (x,xn) (xn,x) + (xn,xn) ?n ,kxn xk 2kxk2 2(x,x) = 0,=xn x0(n ). 9!?MHilbertmH?5fm.TM?k.5f.y3H 3k.5fT,?3MTT ?kTk kTk. y:M?4M?M , T 3MM?.? M 3 xn x0,M 3 yn x0 kTxn Txmk kTkkxn xmk,kTxn Tynk kTkkxn ynk Txn43,46u?%CS?. T(x) = Tx,x M, lim n Txn,x MM, 0,x M 31 o kTxk = kT(y + z)k(y M,z M ,d?) = kTyk = kTyk kTkMkyk kTkMkxk(x H) 11!?THilbertmH?5fkx,y H,(Tx,y) = (x,Ty). yTk.f. y:IyTH?45f. ?xn Hvxn x0,Txn y0, Ky H,d (Txn,y) = (xn,Ty),-n Kk (y0,y) = (x0,Ty) = (Tx0,y) ?y0= Tx0,=T45f,l?d4”nTk 13!?HHilbertm,(x,y)3H H?ux5 ?,uy?5?,3C,?|(x,y)| Ckxkkyk(x,y H).y:3fA B(H),?kx,y H,(x,y) = (Ax,y) kAk = kk, kk =sup kxk=1,kyk=1 |(x,y)|. y: (x,y)uy?5?,?(x,y) uy5?. ?x H,K(x,y)H?k.5,dRiesz Ln, 3x H,?(x,y) = (y,x) =(x,y) = (x,y).N?A : x 7 x, k(x,y) = (x,y) = (Ax,y).du (A(x1+ x2),y) = (x1+ x2,y) = (x1,y) + (x2,y) = (Ax1,y) + (Ax2,y) = (Ax1+ Ax2,y) =A(x1+ x2) = Ax1+ Ax2.q kAxk2= (Ax,Ax) = (x,Ax) kkkxkkyk =kAk kk, A B(H).2dSchwarz?,k |(x,y)| = |(Ax,y)| kAxkkyk kAkkxkkyk ?