1、SKNSK 11. 3N R N (x;y) = (x y)2 d(x;y) =pjx yjy(1) (R; ) m(2) (R;d) m“ymm3uTmlvln“u (x;y) = (x y)2 d(x;y) = pjx yjK5!5w,3R“ n1u (x;y) = (x y)28x;y;z2R(x;z) + (z;y) (x;y) = (x z)2 + (z y)2 (x y)2 = 2(z x)(z y)z 3x;ymu 0=(x;z) + (z;y) (x;y)vn“2ud(x;y) = pjx yj8x;y;z2Rd(x;y) = pjx yj= pjx z +z yj pjx z
2、j+jz yjpjx zj+pjz yj= d(x;z) +d(z;y)vn“ 2. (X; ) mf : 0;+1 ! 0;1 N v8x;y2 0;+1; f(0) = 0; f(x + y) f(x) + f(y)-d(x;y) = f( (x;y)y(X;d) myu8x;y;z2Xdu (X; ) mK1 (x;y) 0 (x;y) = 0,x = y2 (x;y) = (y;x)3 (x;y) (x;z) + (z;y)k f Oe f f(0) = 0x 0k f(x) 0k f(x) 0 f(x) = 0,x = 0“u8x;y;z2Xyd(x;y) vln1d(x;y) =
3、f( (x;y) f(0) = 0 d(x;y) = 0, (x;y) = 0,x = y2d(x;y) = f( (x;y) = f( (y;x) = d(y;x)3d(x;y) = f( (x;y) f( (x;z) + (z;y) f( (x;z) + f( (z;y) =d(x;z) +d(z;y) (X;d) m“ 13. (X;d) my 8x;y;z;w 2 Xk jd(x;z) d(y;w)j d(x;y) +d(z;w)“y8x;y;z;w2Xk d(x;z) d(x;y) + d(y;z) 9 d(y;z) d(y;w) d(z;w)=d(x;z) d(y;w) d(x;y
4、) +d(y;z) d(y;w) d(x;y) +d(z;w)nd(y;w) d(x;z) d(y;x) +d(x;w) d(x;z) d(x;y) +d(z;w)=(J“ K(J d(x;y) x;y Y“ek d(x;x0) !0; d(y;y0)!0Kjd(x;y) d(x0;y0)j d(x;x0) +d(y;y0)!04. N|8X =f(x1;x2; ;xn; )jxi2R;i = 1;2; g8x;y2X d(x;y) =1Pk=112kjxk ykj1+jxk ykjy (X;d) m“yd(x;y)k“=n8x;y;z2XIyd(x;y) d(x;z) +d(z;y)=y1X
5、k=112kjxk ykj1 +jxk ykj 1Xk=112kjxk zkj1 +jxk zkj +1Xk=112kjzk ykj1 +jzk ykjyzk = 1;2; kjxk ykj1 +jxk ykj jxk zkj1 +jxk zkj +jzk ykj1 +jzk ykj=“duf(t) = t=(1 +t) O=jxk ykj1 +jxk ykj jxk zkj1 +jxk zkj+jzk ykj +jzk ykj1 +jxk zkj+jzk ykjjxk zkj1 +jxk zkj+ jzk ykj1 +jzk ykj (X;d) m“ 5*. X(n) 0 1 |nkS|XX
6、(3) =f000;001;010;011;100;101;110;111g2u?x;y2X(n) d(x;y) xy X3X(3)d(110;111) = 1d(010;010) = 0d(010;101) = 3“y (X(n);d) m“y=n8x;y;z2X(n)Iyd(x;y) d(x;z)+d(z;y)“x;y;znd 0!1 |Px = (x1x2 xn);y = (y1y2 yn);z = (z1z2 zn)$d(x;y) = Pni=1(xi yi)“IynXi=1(xi yi) nXi=1(xi zi) +nXi=1(zi yi)yzi = 1;2; ;nk(xi yi)
7、(xi zi) + (zi yi)=“Jwxi;yi 0 1 w“exi;yi O 0!1ziX“ n“ 6. (X;d) mA X A6= ?“y Am8 =Am“y)eAm8zx2A9 0O(x; ) AHA:KkA x2AO(x; ) A=A =x2AO(x; )“(?m8m8Iymm8=“u?mO(x0; )8x2O(x0; ) = ( d(x;x0)=2kO(x; ) O(x0; )“8y2O(x; ) kd(y;x0) d(y;x) +d(x;x0) 09z2Ad(y;A) d(y;z) uz2Akd(x;A) d(x;z)“ujd(x;A) d(y;A)j jd(x;z) (d(
8、y;z) )j d(x;y) + d ?5Q(“ ,ydud(x;A) = infz2Afd(x;z)g infz2Afd(x;y)+d(y;z)g= d(x;y)+infz2Afd(y;z)g= d(x;y)+d(y;A)=d(x;A) d(y;A) d(x;y)“nkd(y;A) d(x;A) d(x;y)“(“ 11. ynmRn:duUI:fxig Rnxi = (x(i)1 ; ;x(i)n ); x0 = (x(0)1 ; ;x(0)n )o limi!1xi = x0 du 8j 2f1;2; ;ngk limi!1x(i)j = x(0)j “ynmRn limi!1xi =
9、x0,8 03N 0iN kd(xi;x0) =nXj=1jx(i)j x(0)j j2!1=209N1 0 ?m;n N1 ok d(xm;xn) 0 ?k;l N2 ok d(yk;yl) Nokd(xm;xn) 09N 0?m;nNokjam anj 09N 0m;nN kd(xm;xn) 09N 0m;nN k (xm;xn) 09N 0n N k d(xn;x0) 09N 0n N k (xn;x0) 03X fyngd(xn;yn) 03NnN kd(yn;y0) N kd(xn;y0) d(xn;yn) +d(yn;y0) 03 N 0 n N n n N k xm 2 Bm B
10、n d(xm;xn) n 03 N 0m;nN d(xm;xn) =1Xi=1jxni xmi jp!1=pN kjxni xmi j N k Pki=1jxni xmi jp NPki=1jxni xijp 0gnNpx = (x1;x2; ;xn);x = (x 1;x 2; ;x n)x NI %a“XJA k.K3 a 0 A I1“ I1 m = 2nNI11; ;I1m7k,I2 := I1k A:“2I2 m NI21; ;I2mk I3 := I2j A:“UYeNSI1 I2 z Ik k A: Ik 3 (pna)=2k+1 4k!1u“KkIk = fx0g8 03kIk
11、 O(x0; )=O(x0; ) kA:l x0 A4:“ 24. (X;d) mA X A6=;eA;8K3x0;y0 2Adiam(A) = d(x0;y0)diam(A) := supx;y2Afd(x;y)g“ydudiam(A) := supx;y2Afd(x;y)g8n 09xn;yn2Adiam(A) 1n 09xn2F1;yn2F2d(F1;F2) d(xn;yn) 0Kd(Ax0;A2x0) 0;9N 0m;nNkkxn xmk1 N kkxn xmk2 Ckxn xmk1 N;s:t:if nN;xn = 0g“y S l15fm4fm“y8x;y2S3N1;N2 n N1
12、;m N2 kxn = 0;ym = 0=3 N = maxfN1;N2gn N xn = 0;yn = 0“u ; 2Kn N x+ y xn + yn = 0“ x+ y2S=Sl15fm“Sl14fm“l1Sx1 = (1=2;0;0; ); x2 = (1=2;1=22;0; ); ;xk = (1=2;1=22; ;1=2k;0; )zxk3k2Nnkxn = 0xk2S“TS4:x0 = (1=2;1=22; ;1=2n; )2l1%uS“ 10. (X;k k1) (Y;k k2) 5Dm- Z = X Yuz = (x;y)2Zkzk= maxfkxk1;kyk2gykzkZ
13、“10y5g5w,“yncky maxfa + b;c + dg maxfa;cg+ maxfb;dgmaxfa+b;c+dg = a+b+c+d+ja+b c dj2a+c+b+d+ja cj+jb dj2= maxfa;cg+ maxfb;dgu8z1 = (x1;y1); z2 = (x2;y2)2Z kkz1 +z2k = maxfkx1 +x2k1;ky1 +y2k2g maxfkx1k1 +kx2k1;ky1k2 +ky2k2gmaxfkx1k1;ky1k2g+ maxfkx2k1;ky2k2g=kz1k+kz2kn“ 11. V0 Lu“N8x = (x1;x2; ;xn; )
14、2 V0kxk= supnjxnj“y V0 l145fm“y8x = (x1;x2; ;xn; )2V0dlimn!1xn = 0=8 0;9N 0n N kjxnj 0y = x“11y(ey = xKkx+yk=kxk+kyk= ( + 1)kxk“)ekx + yk = kxk + kykK (kxk + kyk)2 = (x + y;x + y)=kxk kyk= (x;y)“b3 0 y = xK(y x;y x) = (y;y) 2 (x;y)+ 2(x;x) =kyk2 2kxkkyk+ 2kxk= (kyk kxk)2 0 =kyk=kxk (y x;y x) = 0“g“3
15、 0y = x“ 14. MSmXf8“y1(M?)?)? = M?2M? =(M)?y1k A B )B? A?x2B? )x?B )x?A ) x 2 A?“ x 2 M ) x ? M? ) x 2 (M?)? ) M (M?)?uk(M?)?)? M?“, x2M? )x? (M?)? )x2 (M?)?)?“= k(M?)?)? = M?“2x2M?,x?M,x?M,x2(M)?“ 15. M Hilbert m H 5fm8x2H3M K3y M H 4fm“yduM?4dKn8x2H3)x = y +z;y2M?;z2(M?)?dqkx = x0 +z0;x0 2M;z0 2M?
16、“)ky = z0;z =x0M = (M?)?l M 48“ 16. M Hilbert m H f8y (M?)? M 48“ydu M M 48Iy (M?)? = M“d25K(M?)? = M2d14KM? = (M)?=(“ 18. C 1;1 1;1YmS(f;g) = R1 1f(t)g(t)dt“PC 1;18M8NyM?N9C 1;1 = M N“y8f 2 M;g 2 N= f(t) = f( t);g(t) = g( t)lf( t)g( t) =f(t)g(t)=fg“dum“(f;g) =Z 11f(t)g(t)dt = 0 f ?g= M ?N“8x(t) 2C
17、1;13)x(t) = f(t) + g(t)f(t) = (x(t) x( t)=22M; g(t) = (x(t)+x( t)=22NC 1;1 = M+NqM?NC 1;1 = M N“ 1219. fekgFAm H IOY = spanfengy x2YL x =1Pk=1(x;ek)ek“y)x2YPex =1Pk=1(x;ek)ekKex2Yuv = x ex2Y“du(v;en) = (x 1Xk=1(x;ek)ek;en) = (x;en) (1Xk=1(x;ek)ek;en) = 0;n = 1;2; v?Y2kv2Y v = 0l x = ex =1Pk=1(x;ek)e
18、k“(ex =1Pk=1(x;ek)ekT?-xn =nPk=1(x;ek)ek 2YKkxn!x;(n!1)“x2Y“ 25. Hilbert m H 5fm M 48 =M = (M?)?“y5w,m4“75 kM (M?)?“g?x2(M?)?M 4dKnx = x0 +z;x0 2M;z2M?(z;z) = (x x0;z) = (x;z) (x0;z) = 0z = 0;x = x0 2M= (M?)? M“ 26. fe1;e2; ;en; gHilbertmHIOXff1;f2; ;fn; gH IOXXJP1k=1kek fkk 0;9y = PNk=1 kek2Mkx yk=
19、kx NXk=1kekkN kx xnk2 =kxk2 kxnk2 =kxk2 nXk=1j(x;ek)j2 kxk2 NXk=1j(x;ek)j2 0;9N 0kkP1k=N+1xkk 0;9N 0n N kkSn+1 Snk = kxnk 0, (y;y) (x;y) (y;x) + (x;x) = (y;y) (y;x) (x;x) (x;y) =(y;y) j(x;y)j2(x;x) 0 = (y;x)=(x;x)15, (x;x)(y;y) j(x;y)j2 42. X Smx;y2X kxk=kyk= 1;kx+yk= 2yx = y“ydukx + yk=kxk+kyk= 2d
20、13 K3 0y = x“fw = 1=x = y“ 43. M Hilbert m H 4fm8x2H Kn3M 3 x0 x = x0 + z z2M? P : H !M(x!x0) Kf“y8x2H kkP(x)k2 = (P(x);x)“y8x2H Kn3M 3x0 x = x0 + zz2M?“dP(x) = x0K(P(x);x) = (x0;x0 +z) = (x0;x0) + (x0;z) = (x0;x0) = (P(x);P(x) =kP(x)k245. M Hilbert m H fmy M 3 H =M? =f0g“y)M 3H 8x2H;9xn2Mxn!x;n!1“y
21、2M?Kk(x;y) = limn!1(xn;y) = 0dx?5y 0=M? =f0g“(M? =f0gKk (M?)? = H“q (M?)? = Ml H = MM 3H“ 16SK 31. X5Dmx;y2Xe8f2X kf(x) = f(y)Kx = y“yex6= y Kk x y 6= 0K3 f 2X f(x y) = kx yk =0;kfk= 1f(x y) = f(x) f(y) = 0 g“ 2. Y 5Dm X 45fm8f2X f 3Y fjY = 0 7k f = 0y X = Y“ypp.92-4.2.2 X5DmGXfmx0 2X;d(x0;G) =infy2
22、Gkx0 yk= d 0K73Xk.5fv18x2G;f(x) =02f(x0) = d3kfk= 1“bX6= YK3x0 2X YdY 4d(x0;Y) 0“d3f2X fjY = 0;f(x0) = d(x0;Y) 0 kfk= 1g“ 3. M 5Dm X ?f8x0 X “ox0 2spanM 8f2X e8x2M k f(x) = 0Kf(x0) = 0“y)ex0 2spanM3fxng spanM!x0du8x2Mkf(x) = 0f(xn) = 0-n!1f(x0) = 0“(y“ex0 =2spanMKd(x0;spanM) = d 0d3f2X v8x2spanMf(x)
23、 = 0;f(x0) = d kfk= 1g“ 4. y C 1;1 5f(x) = R0 1x(t)dt R10 x(t)dt2“y8x(t) = C 1;1kx(t)k= 1 jf(x)j=jZ 01x(t)dt Z 10x(t)dtj jZ 01x(t)dtj jZ 10x(t)dtj 2d supkxk=1jf(x(t)j= 2“xn(t) =8:1 t2 1; 1=n)nt t2 1=n;1=n1 t2(1=n;1f(xn(t) =Z 1=n11dt+Z 01=n( nt)dt+Z 1=n0( nt)dt+Z 11=n( 1)dt = 2 1=n!2=kfk= 2“ 5. y5Dm
24、X “5f Yf “m N(f) 3X “y)fYf3x = 0?Y9 0 0:fxng Xxn!0kjf(xn)j 0“?x2Xw,x f(x)xn=f(xn)2N(f)n!1kx f(x)xn=f(xn)!x=x2N(f)“dN(f) 3X“(y“bf YduN(f) 3X =8x2X3fxng N(f)xn!x“l f(x) = limn!1f(xn) = 0g“ 1710. (X;k k1) (Y;k k2) D5mT1 : X ! Y 4fT2 : X!Y k.f“y T1 +T2 4f“y?:fxng Xxn!x9 (T1 +T2)xn!y;(n!1)“duT2k.fK3M 0kT
25、2xn T2xk2 Mkxn xk1 ! 0;(n!1)uT2xn!T2x“l T1xn!y T2x“T1 4fy T2x = T1x=y = (T1 +T2)xT1 +T2 4f“ 12. u?x = (x1;x2; ;xn; )2l1NT(x) = (x1;x2=2; ;xn=n; )y T 5k.f kTk= 1“y8x;y2l19 ; 2KKT( x+ y) = T( x1 + y1;( x2 + y2)=2; ) = Tx+ TyT 5“dkTxk=k(x1;x2; ;xn; )k= supnfjxn=njg supnfjxnjg=kxkkTk 1“x0 = (1;0; )w,kx0
26、k= 1;kTx0k=kx0k= 1“kTk 1T k.5f kTk= 1“ 14. X = Ca;bkxk = maxt2a;bjx(t)jNT : X!X Tx(t) = x(t)y T 5k.f kTk= maxfjaj;jbjg“y5w,“dkTx(t)k=ktx(t)k=jtjkx(t)k maxfjaj;jbjgkx(t)kdTk.5f kTk maxfjaj;jbjgT = supx6=0kTx(t)k=kx(t)k= supt2a;bfjtjg= maxfjaj;jbjg18. XD5m X6=f0gy8x2Xkkxk= supfjf(x)jjf2X ;kfk= 1g“yx =
27、 0 w,“bx6= 08f 2X kfk = 1kjf(x)j kfkkxk = kxkkxk supfjf(x)jjf 2 X ;kfk = 1g“,dn3 f0 2 X jf0(x)j = kxk;kf0k = 1Kkxk = jf0(x)j supfjf(x)jjf2X ;kfk= 1g“ 18jmax(x y;0) max(x z;0)j jy zj?=xyzxz ykj(x y) (x z)j=jz yj=jy zjyxzkj0 (x z)j= x z y z =jy zjz xykj(x y) 0j= x y z y =jy zjyz xz yxkj0 0j jy zjn=(J“
28、Since max(x;y) = 0:5(x y + jx yj); jx + yj jxj + jyj, so jx yj =jx z +z yj jx zj+jz yj i.e. jx yj jx zj jz yj. thenjmax(x y;0) max(x z;0)j = j0:5(x y +jx yj) 0:5(x z +jx zj)j= j0:5(z y) + 0:5(jx yj jx zj)j0:5jz yj+ 0:5(jx yj jx zj)0:5jz yj+ 0:5jz yj=jy zjFZ 11v(x+ )f( )d =Z 11Z 11v(x+ )f( )d e i2 kxdx= Z 11f( )Z 11v(x+ )e i2 kxdxd = Z 11f( )ei2 k Z 11v(y)e i2 kydyd = Z 11f( )ei2 k Fv(k)d = Fv(k)F f(k)19
