TD 6 Espaces vectoriels normés

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TD 6 Espaces vectoriels normés

L:E!E;(x;y)7!(x;2yx)?

E!R+;(x;y)7!max(jxj;jyj)?

kL(x;y)k1=jxj+j2yxj jxj+j2yj+jxj= 2k(x;y)k1: kL(x;y)k1=j 1j+j2 + 1j= 4 = 2k(x;y)k1; kL(x;y)k1= max(jxj;j2yxj)max(jxj;j2yj+jxj) = 2jyj+jxj 3k(x;y)k1: kL(x;y)k1= max(j 1j;j2 + 1j) = 3 = 3k(x;y)k1; ?????kk1:E!R+; f7!supfjf(x)j jx2[0;1]g? ?? ?

L:E!R; f7!Z

f(t)dt: jL(f)j=Z f(t)dtZ jf(t)jdtZ kfk1dt=kfk1: ????N(f)1? n(x) =nx??x1=n

1??x1=n

L(fn) =Z

1=n nxdx+Z

1=n1dx= 112n!n!+11:

1:E!R+;P

k2Na kXk7!P k2Njakj??kk1:E!R+;P k2Na kXk7!supfjakj jk2Ng? k2Na kXk7!P k2N(k+1)ak+1Xk????? j1(P)j=jP(1)j= k2Na k2Njakj=kPk1: k=0Xn? ?? ?kPnk1= 1? ???? j3(P)j=jP(1=2)j= k2Na k2Njakj2 k kPk1X k2N12 k= 2kPk1: k=0Xn? ?? ?kPnk1= 1??3(Pn) = kPk1= maxfjanjg X n2Njanj=kPk11: k=0Xn? ?? ? ????(E;kk1)?

0jPj> Cn?

1:nX k=1a kXk7!max0knjakj

N:P7!Z

jPj n;nP k=0x n2N? k=0x k=0kxkk

E? ????? ?? ?????(Pn

8" >0;9N2N:8p > qN;pX

k=0kxkk qX k=0kxkk=pX k=q+1kxkk< ": k=0x kqX k=0x k=q+1x k=q+1kxkk< ": j1?xj=yNj+1yNj? ????? k=0x

Nk=yn0+nX

k=1y

Nk+1yNk=yNn+1:

k=0kxNkk nX k=02 k= 2: zwx ? ????? ??x?PC(x)?

2kuk2+ 2kvk2=ku+vk2+kuvk2:

ku+vk2+kuvk2=kuk2+ 2hu;vi+kvk2+kuk22hu;vi+kvk2= 2kuk2+ 2kvk2; "=fy2Cj kxyk d(x;C) +"g:

2kxyk2+ 2kxzk2=kyzk2+k2(xw)k2;

kyzk2= 2kxyk2+ 2kxzk24kxwk2; kyzk22(d(x;C) +")2+ 2(d(x;C) +")24d(x;C)2="(8d(x;C) + 4"): ??n= 1? ???A=PTP1? R=01 1 0quotesdbs_dbs7.pdfusesText_5

[PDF] TD 6 Espaces vectoriels normés