天津大学应用泛函分析课后题答案.pdf

8 0.11?SK) 1 1.21?SK) 5 2.31n?SK) 12 3.41o?SK) 17 i 1!1?SK)1 0.1 1?SK) !?KKK 1g,8?88( ) 20,1 ?88( ) 38?f88( ) 4k8?88( ) 5kn88( ) 6. Rnd?NknI:?8 Qn?.( ) 7m a,b ?knX?N8( ) 8?p?mm|?88( ) 9. 5m8 X K ?N?/0/ 0 vA?A5?o?nN.( ) 10. ? M1, M25m X ?fm, K M1 M2 X ?fm.( ) 11. ? M1, M25m X ?fm, K M1 M2 X ?fm.( ) 12. 5m Pna,b n ?.( ) 13 Mmn(R) mn 5m( ) 145f?m5m( ) 155?kmk?( ) 16Sm5m?vSn?nN( ) 17. ?Sm (X,(,) ,Kk |(x,y)| (x, x)p(y,y) . ( ) 18. 3Smk |x + y|2+ |x y|2= 2(|x|2+ |y|2) .( ) 19. ? X ?Sm, x,y X , K xy |x + y|2= |x|2+ |y|2.( ) 20. ? X Sm, A X , K A X ?fm.( ) 21. ? x1, x2, , xn Sm X ?X, K x1, x2, , xn 58. ( ) 22Sm5| x1, x2, , xn L?z?X ( ) ?!WWWKKK 1. ? A 5m X ?f8,K sapnM A ?fm. 2. ? M1, M25m X ?fm,K M1 M2? ( M1 M2= ) . 3. ?X5mXnm? (3n?5?n + 1?5) 28 4. ? X ESm, o4z? x,y X (x,y) = 1 4|x + y| 2 |x y|2+ i|x + iy|2 i|x iy|2. 5. ? X Sm, x X , e u X k (x,u) = 0 ,K x = . 6. ? A Sm X ?f8, A ,K A A= . 7. ? e1,e2, ,en Sm X ?IO?X,K x spane1,e2, ,en , k x = n P k=1(x,e k)ek. 8. ? X,Y K ?5m, T : X Y 5f,K T() = . 9. ?kN? f : A B ,e R(f) = B ,K f ()?. nnn!yyyOOO 1. ? 1 p q + y5mkX p q. yyy 5?XJx (0,1),Kkxq xp. ?x = n p,du?P n |n|p N|n| 0 .() 34. D5m?48?8() ?!WWWKKK 1. ? X kDm,o X ?k(?d?). 2. ? X D5m, Y?:(lim n |xn| = | lim n xn|) 3. ? X Banachm, Y X ?fm, o Y Banachm? (Y4fm ) 4. ? A 5m X ?f8,K spanA A ?fm. 5. ? A D5m X ?f8,K A A ?48. 6. ? A Banachm X ?4fm,K A D5m Banachm . 7. ? kk,kk5m X ?d, xn X ?S?, x X . e xn k ku x , K lim n kxn xk= 0 . 8. ? X 5m, d X ?, o d U? d(x + z,y + z) = d(x,y), d(x,y) = |d(x,y). 9. ? X D5m, | | X ?, o X Sm, | | S p?7 | v (|x + y|2+ |x y|2= 2(|x|2+ |y|2) (1o /). 10. X,Y m, f X ? Y ?N?,o f X YN? ? Y ?m8 G , f1(G) X ?m8 nnn!JJJKKK 1. e?D5m?( c ) (a). (Mnn(C),kk) , kAk = n P i=1 n P j=1 ? ? ?a ij ? ? ?, (A = h aij i Mnn(C) . (b). (,kk) , kxk = sup iN |i|,(x = i ) . (c). (Ca,b,kk2) , kxk2= ( b R a |x(t)|2dt)1/2,(x Ca,b) . (d). (Cn,kk) , kxk= max 1in |i|,(x = (1,2, ,n) Cn) . 2. e?D5m?( b ) 1?!1?SK)7 (a). (Mnn(C),k k) , kAk = n P i=1 n P j=1 |aij| (A = (aij) Mnn(C). (b). (,k k) , kxk = sup iN |i|, (x = (1,2,i,) ) . (c). (Ca,b,kk2) , kxk2= b R a |x(t)|2dt 1 2 ,(x Ca,b) . (d). (Cn,kk) , kxk= max 1in |i|, (x = (1,2, ,n) Cn) . 3. ? X D5m, A X , K A 48?( d ) (a). A m8.(b) A A . (c). A k8.(d) A ?S?43 A . 4. ? X D5m, A X , x X .Ke?(?(?( d ) (a). A X ?48,?=? A A . (b). A X ?kfm,K A X ?4fm. (c). x A ?=?3 xn A ,? lim n xn= x . (d). x A ?=?3, r0 0 ,? B(x,r0) A , . 5. ? X Banachm, A X . K A ;8,?=?( d ) (a). A ?;8.(b). A 48. (c). A k.48.(d). A ?;?48. 6? 1= a = an ? ? ? P n=1 |an| g) 8 yyy ? R, ef(x) g(x) , K f + g. dkn?5, z x E,3knr? f(x) r + g(x) u E(f g + ) = rQ E(f r) E(g r)E(g r) E(g r) E(g g + )8. AO/, = 0, E(f g) 8? 5. y f(x) E ?kn r , 8 E(f r) XJ8 E(f = r) , f(x) yyy ? R,3N?kn?rn?rn rn)8,Kd8?$5 E(f ) = n=1 E(f rn) 8. 8 E(f = r) , v?y f(x) ? 6. ? f(x),g(x) E , g(x) , A? f (x) g(x) , f (x) ) ?v?yf?5,XE = R+, f(x) = ex, g(x) = ex ,f(x) g(x),R 0 exdx = 1,? R E f(x)dx = .? 7. ? f(x) 0 , - f(x)n= f(x),f(x) n 0,f(x) n. K? f(x) A?k, k lim n Z b a f(x)ndx = Z b a f(x)dx yyy pR b a f(x)dx n),E(f n) E(f n + 1) 148 8?5, lim n mE(f n) = 0. Z b a f(x)dx = Z E(fn) f(x)dx + Z E(fn) f(x)dx ?Y5?y lim n R E(fn) f(x)dx = 0, lim n Z b a fndx = lim n Z E(fn) f(x)dx = Z b a f(x)dx. ? 8. ? f(x) a,b , XJ?k. (x) , k Z b a f(x)(x)dx = 0, K f = 0(a.e.) yyy du f(x) a,b ,mE(f = ) = 09lim n R E(fn) f(x)dx = 0. ? f(x)n= f(x),f(x) n 0,f(x) n. fnk.,b?(x) = sign(fn(x), (x)k., k Z b a f(x)(x)dx = Z b a |f(x)n|dx = 0, Lf3E(f n)A?, -n , f(x) = 0,(a.e.).? 9. 4 lim n (R) Z 1 0 nx 1 2 1 + n2x2 sin5nxdx. yyy due?1 + x2 2x, ? ? ? ? ? ? ? nx 1 2 1 + n2x2 sin5nx ? ? ? ? ? ? ? nx 1 2 2nx 1 2 x kR 1 0 1 2 xdx = 1. z?x 0,1, lim n nx 1 2 1 + n2x2 = 0. Ank lim n (R) Z 1 0 nx 1 2 1 + n2x2 sin5nxdx = 0. ? 1n!1n?SK)15 10. ? p,q,r v 1 p + 1 q + 1 r = 1 ?n?, K?k. f, g, h, k Z b a |fgh|dx |f|p|g|q|h|r. yyy du1 p + 1 q + 1 r = 1 P = qr q+r, 1 p + 1 = 1 , A H older? Z b a |fgh|dx |f|p|gh|. |gh|= Z b a |gh|dx !1 = Z b a |g| qr q+r|h| qr q+rdx !1 . Pp0= q+r r ,q0= q+r q ,2gAH older? Z b a |g| q p0|h| r q0dx Z b a |g|qdx ! r q+r Z b a |h|rdx ! q q+r . “=? Z b a |fgh|dx |f|p|g|q|h|r. ? 11. ?M(E)kE|?8, U?5$, M(E)5 m yyy f,g M(E), R, (f + g)(x) := f(x) + g(x),( f)(x) := f(x) ?5Lf + g M(E), f M(E). 5f(x) = 0,(a.e)?, 5f(x) = g(x)(a.e)?f = g. du?5$5 ovM(E)5m? 12?y L2a,b , L(E)M(E)?5fm. yyy L2a,b M(E), ?f,g L2a,b, K, f +g L2a,b,f L2a,b, L2a,bM(E)?5fm La,b M(E), ?f,g La,b, K, f + g L2a,b,f La,b, La,bM(E)?5fm? 13. ? E m 0,2 ?n8, OLebesgue R E sin ? x 2n ? dx , 4 lim n Z E sin ? x 2n ? dx ) pOREsin ? x 2n ? dx , |kn8? R Q sin ? x 2n ? dx = 0 9 ?5, Z E sin ? x 2n ? dx = Z E sin ? x 2n ? dx + Z Q sin ? x 2n ? dx = Z 2 0 sin ? x 2n ? dx = 2n(1 cos( 2 2n ) 168 lim n Z E sin ? x 2n ? dx = 0. 1o!1o?SK)17 3.4 1o?SK) !?KKK 1D5m?k.5fY5f( ) 2D5m?5fk.?( ) 3kD5m?5fk.5f( ) 4. ? X,Y D5m, e Y k?,K B(X,Y) ?.( ) 5?m?a?( ) 6. ? X D5m, T : X X, S : X X 5f, K S T : X X 5f.( ) 7. ? X,Y D5m, N? T : X Y ?m N(T) X ?5fm. ( ) 8. X 5m, f X ?5,o N(f) X ?.( ) 9. ? X,Y D5m, T : X Y k.5f, K N(T) 4fm.( ) 10. ? X,Y D5m, 5f T : (X,kk) (Y,kk) Y?,?= ? T(S(,1) Y ?k.8.( ) 11. ? X,Y D5m, T : (X,kk) (Y,kk) 5f,K T k.?= ? T 3: X Y.( ) 12. d A Mnn(C) (?l Cn? Cn?5fY?.( ) 13. ? X,Y D5m, T : (X,kk) (Y,kk) Y, e M X ;8, K T(M) Y ?;8.( ) 14. ? X D5m, T,S B(X,X) ,K kTSk kTkkSk .( ) 15. ? X , Y D5m, X k?,KN? T : X Y Y?.( ) 16. ? X,Y D5m, T : (X,kk) (Y,kk) k.5f, e Y k ?,K T(B(,1) Y ?;8.( ) 17. ? T : 2 2?: x = (1,2, ,n,) 2, Tx = (1 1 , 2 2 , , n n ,) ,K T : 2 2k.5f.( ) ?!WWWKKK 1. ? X,Y D5m, f : (X,kk) (Y,kk) V?,K ? ? ?f 1 f ? ? ? = 1 . 2. ? X,Y D5m, f : (X,kk) (Y,kk) YN?,e lim n xn= x0 X f(x0) = y0, K lim n f(xn) = y0. 3. ? C1,2 ? kfk = max 1x2 |f(x)| ( f C1,2) . k.5f T : C1,2 C1,2 ?: (T f)(x) = Z x 1 f(t)dt,f C1,2, 188 K kTk = 3. 4. ? X,Y D5m, T : X Y k.5f,K T ?A?d / |Tk| = infr 0; ? ? ? |Tx|Y r|x| = sup x, |Tx|Y |x|X ! = ( sup |x|=1| |Tx|Y) = (sup |x|1 |Tx|Y) 5. ?5m Ca,b ? kxk = max atb |x(t)| ( x Ca,b) . k.5 f : Ca,b R ?: f(x) = Z b a x(t)dt,(x Ca,b), K kfk = b a . 6? X D5m, f : X C k.5, K f ?A/ |f| = infr 0; ? ? ? |f(x)| r|x| = sup x, |f(x)| |x| ! = (sup |x|=1 |f(x)|) = (sup |x|1 |f(x)|) 7. ? X D5m, Km Xo Banach m nnn!JJJKKK 1? X,Y K ?D5m, e X k?, K( a ) (a). 5f T : X Y k.?.(b). B(X,Y) ?. (c). Y ?d?.(d). X ?f8k.?. 2. ? X,Y D5m, T : (X,kk) (Y,kk) 5f, K T Y? ( b ) (a). T(X) Y ?k.8.(b). T 3 X ?,:Y. (c). T ?.(d). T V?. 3. e?5fk.5f?( c ) (a). A Mnn(C) , 3m (Cn,| |1) f Ax = n X i=1 a1ixi, n X i=1 a2ixi, , n X i=1 anixi , (x = xi Cn). (b). (,kk) , kxk = sup iN |i|,(x = (1,2,n,) l) . f A : 2 Ax = ? 1, 2 2 , 3 3 , , n n , ? (c). (Ca,b,kk),kxk= max atb |x(t)|,(x Ca,b). q(t) Ca,b Ax(t) = x0(t) + q(t)x(t) 1o!1o?SK)19 (d). (Ca,b,kk) , kxk= max atb |x(t)|,(x Ca,b) . q(t) Ca,b Ax(t) = q(t)x a + b 2 ! . 4. e?=?( c ) (a). A Mnn(C) , kAk = max 1in n P j=1 |aij|,(A = (aij) Mnn(C) . (b). A Mnn(C) , kAk = n P i=1 n P j=1 ? ? ?a ij ? ? ?2 !1/2 (A = ? aij ? Mnn(C) . (c). A Mnn(C) , kAk = max 1i,jn |aij|,(A = ? aij ? Mnn(C) . (d). A Mnn(C) , kAk = max 1jn n P i=1 |aij|,(A = ? aij ? Mnn(C) . ooo!yyyKKK 1. 3m (2,| |2) , f T : 2 2 Tn = n+1