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n = (k)k2N 0 W kfk1= sup x2[0;1]jf(x)jkfk2=
Z1
0jf(x)j2x
1 2 ()0 C([0;1])
ĕ n >0 (ak)1kn(bk)1kn
(i;j)2[1;n]2;ai+bj6= 0 m 0< mn Dm= 1 a1+b1 1 a1+b2:::1 a1+bm1 a2+b1 1 a2+b2:::1 a2+bm 1 am+b1 1 am+b2:::1 am+bm
R(X) =
n1Q k=1 (Xak) nQ k=1 (X+bk):
R(X) R(X) =
nP k=1 ckX+bk ck cnDn=R(an)Dn1
Dn ĕ
R(a1)
R(a2) R(an) Dn= Q
1i Q 1in 1jn (ai+bj): AE d(x;A)
d(x;A) = infy2Akxyk d(x;A) = 0 x (An)n0 E A=S n0And(x;A) = limn!1d(x;An) ĕ V E B=y:kyxk kxk:
B\V d(x;V) =d(x;B\V) x2E
x2E y2V d(x;V) =kxyk <;>Ekxk=p< x;x > EV? VV?=E8(x;y)2VV?;< x;y >= 0
x2E y2Vn f(x)g kxyk>kx(x)k d(x;V) =kxyk (x1;x2;:::;xn)2En G(x1;x2;:::;xn) M(x1;x2;:::;xn) =
< x1;x1> < x1;x2>< x1;xn>
< x2;x1> < x2;x2>< x2;xn> < xn;x1> < xn;x2>< xn;xn> G(x1;x2;:::;xn) = 0 (x1;x2;:::;xn)
x2E d(x;V)2=G(x1;x2;:::;xn;x) G(x1;x2;:::;xn):
ě k:k1k:k2
AC([0;1]) A1A2 A k:k1k:k2
f2C([0;1]) d(f;A) fA k:k1 f2C([0;1])k:k2(f) k:k1(f) AC([0;1]) A1A2
02V0 2 k:k1 m0; m2V1 m0; m2V2 ě ĕ W k:k2
0limn!+1d(;Wn) = 0
0 d(;Wn) =1p2+ 1 nY k=0 jkj k++ 1 0 k++ 1 k2N (k)k2N +1 x2[0;]7!x x++ 1 n ě ĕ W k:k1
W C([0;1]) k:k1 P1
n =Pn k=0akk Wn k1 k2 f0;1;:::;ng 1 k( )k1 k1 nX k=0 akkk1k2: (k)k2N ¨(i) :0= 0
(ii) :k1 k1: P1 n W C([0;1]) k:k1 (ii) (ii0) : infk1k>0 n sh RR un= nX k=1 pn(k2+n2) n (un) f R f kfk1= sup x2[0;1]jf(x)jkfk2= Z1
0jf(x)j2x
1 2 §xax2]0;1]
0x= 0 p2(x) =x p2 0 [0;1]
ĕ = (k)k2N
((k)k2N) E k:k2 k:k1 f2E (Pn)N limn!+1kfPnk1= 0 (f) :x7!xf0(x): a:a7!xa (k)k2N (k)k2N E f02Ek:k E f0 AE d(f0;A) d(f0;A) = infa2Akf0ak dA:f7!d(f;A) d(f0;A) = 0 f0 A (An)n0 E A=S n0An (d(f0;An))N 8" >0;9N2N:d(f0;AN)d(f0;A) +"
limn!+1d(f0;An) =d(f0;A) F E
B=u2E;kf0uk kf0k:
B\F F d(f0;F) =d(f0;B\F)
ě N1N2
AE A1A2 A k:k1k:k2
f2E d(f;A) fA k:k1 0jf(x)j2x1
2 E f2Ekfk2 kfk1 AE A1A2 (x7!1(1x)n)N 0:x7!1 k:k2 k:k2k:k1 k2N; k:x7!xk G1 ĕ k2N; k2G2 ĕ n >0 (ak)1kn(bk)1kn
Dm= 1 a1+b1 1 a1+b2:::1 a1+bm1 a2+b1 1 a2+b2:::1 a2+bm 1 am+b1 1 am+b2:::1 am+bm R(X) =
n1Q k=1 (Xak) nQ k=1 (X+bk): R(X) R(X) =
nP k=1 ckX+bk ck cnDn=R(an)Dn1: Dn ĕ
R(a1)
R(a2) R(an) Dn= Q 1i Q 1in 1jn (ai+bj): ě ĕ F k:k2 k:k1
f02E = (k)k2N (k)0kn m0limn!+1d(m;Fn) = 0 Z1 0jf(x)j jg(x)jx
E ij < i;j> (e1;e2;:::;ep) E G(e1;e2;:::;ep) G(e1;e2;:::;ep) =
< e1;e1> < e1;e2>< e1;ep> < e2;e1> < e2;e2>< e2;ep> < en;e1> < en;e2>< ep;ep> m d(m;Fn)2=G(1;2;:::;n;m) G(1;2;:::;n)
m0d(m;Fn) =1p2m+ 1 nY k=0 jkmj k+m+ 1: m0 k+m+ 1 k2N (k)k2N +1 x7!mx x+m+ 1[0;m] n F E k:k1 P1
nquotesdbs_dbs43.pdfusesText_43
Sujets et corrigés des DS de mathématiques et d'informatique