[PDF] Université Paul Sabatier L3 MAF 2015-2016 Topologie et analyse

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i2IEi=fxj 9i2I;x2Eig??\i2IEi=fxj 8i2I;x2Eig: P(E)? n2NDn= ??A?? ??? ???" >0????K?? ?????? ???????m??? ???mx < m+ 1? ?1)5? ??1=n!`?????1n(n+1)=1n

1n+1!``= 0? ???? ????" >0????K? ?? ?????? ????

n2N??? ???? ? ???? ??? ??????? ? ?????? ???? ??????? ????? ??? ???1n(n+1)< "? ???? ???? ???? ?????? ???? ??????? ?? ???? ????? ???????m? ??? ?????? ????mx < m+ 1: q >1=(ba) etp1aq < p; q >0 etaq < paq+ 1< bqdonca < p=q < b: ?3)2? ??Q??? ????? ?????? ???? ????" >0????K? ?? ?????? ??? ???????q >0??p???? ???

0< p=q < "? ????8nq;0<1=n1=qp=q < "? ?? ??? ?????? ???1=n!0?

?=r? ??? ???a < s < b? ?]a;b[ :=fx2Eja < x < bg? ?fx2Ejx < bg? ?fx2Eja < xg? ?E? ??????? ??E??? ?????? ??X?? ??????? ??? ?

Y=RX=R?Y=N[ f+1g???Y=f1=njn2Ng [ f0g?X=R?

???? ?????? ?? ??????U??? ???a2U??UV?? a?? ?????? ????U? ?? ???????O?? ??? ??????? ??? ?? ??????? ??U??? ???? ? ??? ?????? ?OU ??? ???Oa???? ?????? ????U? ??UO???8a2U?a2Oa??OaO? AA?

A\V6 fxg?

?????(1)n?

8V2 V(x)8N2NfunjnNg \V6=?

8N2Nx2funjnNg:

?E??????? ??? ?????? ??E?

8X;YE:XX;X=X;X[Y=X[Yet?=?:

Fr(A) :=AnA=A\EnA;

??????? ??F? F ???X????B??? ??? ?????

UV????U?????? ??X??V?????? ??Y?

??? ????? ????X??B????? ????Y?????AB??? ????? ????XY? ? ??X??Y???? ??????? ?????XY?????? ??????? ??? ????? ?? ???????XY?

W??a??? ???f(W\A)V?

?? ??Y??? ?????? ?? ?? ??? ????? ?????? ?????? ????? ???? ??? ??????? ?? ???? ????? ??????A??X?f(A)f(A)? ?? ???? ????? ??????B??Y?f

1(B)f1(B):

A=f(1=n;y)jn2N;y2[0;1]g;

B=f(x;arctanx)jx2Rg;

C=f(x;y)2R2j1< x2=3 +y2=5<2 ety <3x4g:

???T0? f(A)B,Af1(B): ??????? ????? ?????gf?????? ????E? ?? ?????a? g(un)!g(`):

X??X fyg?

?????? ??? ?? ????? ??X? XY? ????? ???? ??????x?y??z??E? ??y6=x? d

1(f;g) = min(1;e1(f;g));ou e1(f;g) = sup

x2Xd(f(x);g(x))2[0;+1]: p?? ??????? ??? k(x1;:::;xn)k1= max(jx1j;:::;jxnj): B(a;r) =fx2Ejd(x;a)< rg; B0(a;r) =fx2Ejd(x;a)rg; S(a;r) =fx2Ejd(x;a) =rg: ;R)? ??????? ????? ???? ???rRd( ???? ???2rRd( ;a)???? ?? ???????a2B(!;r)? ??a? diam(A) = sup x;y2Ad(x;y)2[0;+1]: ? ??????? ???diam(A) = diam(A)? ? ???? ????x2E?A??? ?????? ???? ??? ????? ?? ??????x? jd(x;z)d(y;z)j d(x;y); ???? ?? ??????? ????k0? ??

8x;y2E; dF(f(x);f(y))kdE(x;y):

x7!d(x;A) = infa2Ad(x;a);

A??B????R2? ???? ????R? ???? ???d(A;B) = 0?

z}|{

B0(a;r)B(a;r)

x2A,d(x;A) = 0:

A????? ???d(x;an)!0:

8" >0;9 >0;8x;y2E; dE(x;y)< )dF(f(x);f(y))< ":

k k d((x;y);(x0;y0)) = max(dX(x;x0);dY(y;y0)) ??X??V?????? ??Y???? ???(x;y)2UV??UVO? ???? ?? ??????r1;r2>0???? ??? B dX(x;r1)BdY(y;r2)O????? ?? ??????r= min(r1;r2)?Bd((x;y);r)O? d p((x;y);(x0;y0)) =k(dX(x;x0);dY(y;y0))kp: kXdX(x;x0) +kYdY(y;y0)(kX+kY)d((x;y);(x0;y0))?

8" >0;9N2N;8p;qN; d(up;uq)";

limn!1sup mnd(um;un) = 0:

8p;qN;d(up;uq)", 8mnN;d(um;un)", 8nN;sup

mnd(um;un)": ????u??? ????? ?? ??????? ?? ?? ??????N??? ???8mN?d(um;uN)1???? ?? ????? ??? ?????? ??? ?????? ?? ????N? ??? ???8p;qN;d(up;uq)"??9qN??? ???d(uq;`)< "????8pN;d(up;`)<2"? jx3y3j?? ?? ?????? ?????q! 1???? ?? ??? ???x2X?8"2]0;1[;9N;8pN;d1(fp(x);f(x))";?? ??? ?????? ???fn!f? ?? ??????A????? ????E??????? ??a??? ????? ?? ?????? ????A? ?????a??? ??? ????? ??

X??Y?? ?????

??? ????F=f`g? ?? ???? ????? ????? ?? ??????(xn)n????E? ?? ?????(f(xn))n????F??? ?? ??????? g:E!F??? ?g(x)??? ?? ?????? ??(f(an))n? ???? ????? ?????(an)n????A?? ??????x? ?? d(an;xn)??d(f(an);g(xn))??????? ????0???? ??????? ???? ??????n?? ??????? ????an? ???? ??? ??????? ??E: ?????E???? ???^E???? ??????? ? ?? ????? ??? ??f??? ?? ???????x??? ???f(x) =x: x k

8mn;d(xn;xm)X

ni8n2N; xn=2 [k

8(i;j)2N2;i6=j)d(xi;xj)"

f

K:=f(x;y)2XXjd(f(x);f(y))"g

?"? ????jf(x)g(x)j< ":

8u2]1;1[p1u= 11X

n=1a nun????an=12 (112 )(212 ):::(n112 )n!: ?????? ???PN(u) := 1PN ???? ???? ??????N >0?? ????u2[0;1[? N X n=1a nun1X n=1a nun= 1p1u PN n=1anun<1???? ?????N???? ?? ???????u!1?PN n=1an1? ??? ??? ???A?????? ??? ?????? ??X?? ???? ????? ?????fx;yg X? ?? ??????f2A????? ??? ?? ???? ?????? ????? ??? ???? ??X??? ???? ?X? V

0=U\C? ?? ?U0\V

0=???V0\U

0=?? ?? ?????? ?????U00=fx2Xjd(x;U0)< d(x;V0)g??

V R?

R??? ?? ?????? ?? ????? ??? ??? ???????0??1?

??C:

QQ?(QR)[(RQ)?(QQ)[[(RnQ)(RnQ)]?

?? ???? ? ?? ???? ????? ??X???? ???? ????? ????? ? ?? ????? ???x0: ??????C??? ???? ??????? ??8x;y2C;[x;y] :=ftx+(1t)yjt2[0;1]g C?? ?? ???? ????? ?Rn????? ???? ????? ??? ??????? ???? ????? ??n >1????? ??? ??n= 1? ????R????? ?????? ?????? ? ?????Rn????n >1?? ?????X??? ??????? ??? ????? kfk1= sup x2Xkf(x)k: kfka:=kfk1+ sup x6=ykf(x)f(y)kd(x;y)a: ??E??? ?? ?????? ? ??? ???????E=R? ?????C(X;E)0;a?????? ?kxk= 0,x= 0E? ?jkxk kykj kxyk? ?????EF????? ???? ??? ????? ????? ???? ??????? ????? ?????? ???k(x;y)k= max(kxkE;kykF)?? ??????(x;y);(x0;y0)2EE??"=k(x0;y0)(x;y)kEE= max((kx0xk;ky0yk)? k(x0+y0)(x+y)k kx0xk+ky0yk 2" ??????(;x);(;y)2RE??"=k(;y)(;x)kRE= max((jj;kyxk)? ?? ????M?? kyxk k()yk+k(yx)k 2M" E? FF: z+UC???? ???? ????u2U?z+u=t(x+u) + (1t)(y+u)2[x+u;y+u]C?? ??u??? ?????? ??? ?? ????? ?????B0(0;1)? (2)3)? ???? >0??? ???kxk ) ku(x)k 1:??????kxk 1) ku(x)k 1=: (3)5)? ????M??? ???kxk 1) ku(x)k M:?????ku(y)u(x)k Mkyxk: (3,4)? ????M??? ???kxk 1) ku(x)k M:????? ???? ???? ????R?kxk R) ku(x)k MR:

8x2E;ku(x)kFMkxkE:

jjjujjj= sup x2E;x6=0ku(x)kFkxkE= sup x2E;kxkE=1ku(x)kF= sup x2E;kxkE1ku(x)kF:

9; >0; N1N2N1:

8a= (ak)k2`1u(a) =1X

k=0 kak k=0kk1jakj= kk1kak1? ???jknj ! kk1? fv2Gjv(z) =v(x) +v(y)g???? ???? ???(x;y;z;;)2S3K2???? ???z=x+y? ?f??? ?????? ??? ?? ??????? ??? ?????? ????? ???Ei? ? ?? ?????? ?? ????M0??? ???

8xi2Ei;kf(x1;:::;xn)kFMkx1kE1:::kxnkEn:

?????jjjfjjj? ??? ?? ???? ????? ????M0??? ???

8xi2Ei;kf(x1;:::;xn)kFMkx1kE1:::kxnkEn:

jjjfjjj= sup x i2Ei;xi6=0kf(x1;:::;xn)kFkx1kE1:::kxnkEn= sup x i2Ei;kxikEi=1kf(x1;:::;xn)kF= sup x i2Ei;kxikEi1kf(x1;:::;xn)kF:

L(E1;:::;En;F)???? ?? ????? ????? ??????

ju(xn)j !+1? ? ?????? ???? ??????? ????N?u(xn)6= 0? ?? ??? ?????? ?? ??????yn:=eu(e)u(xn)xn? f:HR!E;(x;)7!x+e: ???v????? ????? ???8x2E;v(x) =xu(x)e? S2E??? ????? ??????? ?? ????? ?? ?? ????? ?? ?????P+1 n=0xn? ? ??? ?????P1 n=0xn?????E? ??P1 ? ??P+1 n=0un=U??P+1 n=0vn=V?????P+1 n=0(un+vn) =U+V? +1X n=0u(xn) =u +1X n=0x n! ?? ?????kPn k=0u(xk)u(S)k=ku(Pn k=0xkS)k jjjujjjkPn k=0xkSk? lim n!1sup mn X n??S0 (n)< `? ??? ??????? ?????? ???? ????`???? ??????? ?S0'(n)!`????? ???S0'(n)1` < S0'(n)??S0'(n)S0'(n)1=x('(n))!0??

n2Nxn??? ???? ??P1 n=0xn?? nxn ??????? ???? ????" >0? ?? ?????? ??? ?????? ????J"??N????? ??? ???? ????? ?????? ????K??N P k2Kxk ???? ?? ??????? ????p >0? ?? ???kP i2JxikpP i2Jkxikp k=0xk??S0n?????? ??S0=P1 j=0x(j)? ???? ????" >0?? ????nJ"[1(J")? ?????? K

1=f(0);:::;(n)g n f0;:::;ng??K2=f0;:::;ng n f(0);:::;(n)g:

kS0nSnk= X k2K1x kX k2K2x k X k2K1x k X k2K2x k <2";

8" >0kS0Sk 2";

??????? ??????? ?? ?????? ?? ?? ????A??? ?? ?????? ??? ???8a;b2A;kabk kakkbk: (A;+;)? ??? ???(ab) = (a)b=a(b)? ??E??? ??????? ?????L(E)?????? knuk=P knuk(1Au) = 1Aun+1: ???? ???khk<1=ka1k: ????K=R??C? hx;y+zi=hx;yi+hx;zi? kx+yk2=kxk2+kyk2+ 2Re(hx;yi): x?y? ??hx;yi= 0? hx;yiR:= Re(hx;yi): ????h;iR? hx;yi=hx;yiR+ ihx;iyiR; ???????Re(hx;iyi) = Re(ihx;yi) = Im(hx;yi)? hx;yi=nX k=1x ky k; n=0jxnj2<+1?? hx;yi=X n2Nx ny n hf;gi=Z fgd: hx;yi=X

1i;jnx

iMi;jyj=xTMy hx;yi=xTPDPTy=uTDv=nX i=1 iuivi i=1iuivi= 0???v= 0? jhx;yij kxkkyk: ? ??kyk 6= 0? ????=thx;yi????t= 1=kyk2? ?????? kxyk2=kxk22Re(hx;yi) +jj2kyk2 =kxk22tjhx;yij2+t2jhx;yij2kyk2 =kxk2 jhx;yij2=kyk2

8t2Rkxk22tjhx;yij20

kx+yk kxk+kyk ? ??K=R?hx;yi=12 (kx+yk2 kxk2 kyk2) =14 (kx+yk2 kxyk2)? ? ??K=C?hx;yi=14 P 3 k=0ik x+ iky 2?? kx+yk2+kxyk2= 2kxk2+ 2kyk2

BC2+ 2AI2??I??? ??

kxk2+kyk2=12 kxyk2+ 2 x+y2 2 14 kx+yk2 kxyk2?

8x;y2Ehx;yi:=kx+yk2 kxyk24

?hx;0i= 0? ?hx;y+zi= 2hx=2;yi+ 2hx=2;zi???

2hx=2;yi+ 2hx=2;zi=12

kx=2 +yk2 kx=2yk2+kx=2 +zk2 kx=2zk2 12 kx=2 +yk2+kx=2 +zk2 kx=2yk2 kx=2zk2 14 kx+y+zk2+kyzk2 kxyzk2 kyzk2 14 kx+y+zk2 kxyzk2 =hx;y+zi;

8x;y2Ehx;yi:=hx;yiR+ ihx;iyiR????hx;yiR:=kx+yk2 kxyk24

8x;y2Ehix;yi= ihx;yi??hy;xi=hx;yi;

hix;yiR=hx;iyiRethix;iyi=hx;yiR: ????? ?????kPn i=1xik2=Pn i=1kxik2: ??xy?F? d(x;y)? ????d(x;F) =d(x;y):

8z2F2hxy;zi+kzk20:

8u2Fhxy;ui= 0:

?? ? ???? ????xy?F? y=nX i=1hx;eiiei: i=1jhx;eiij2? ??? ????? ????E?

8x2E;X

i2Ijhx;eiij2 kxk2; ???F? p

F(x) =X

i2Ihx;eiiei: ??x2F?? ? ????x=P

8J???I;X

i2Jjhx;eiij2 kxk2????X i2Ijhx;eiij2 kxk2: y=X i2Ihx;eiiei ?? ????P kerpF=F?etE=FF?: (G?)?=G: ????G?=G ?=E?=f0g? ???? ??????u??h;b=kbk2i??? ???? ????? ?? ???? ?????? ??b???? ????? ???? ???????

1k ? ???? ????b2B?P

A(b) :=fa2Aja6?bg: card(A(b))card(N):

A[ b2BA(b): ? ?? ??? ???(E;)??? ? P(X)n f?g !X????? ??? ????? ????? ?????? ??? ????Y??X?f(Y)2Y? ??? ????? ?????? ???P(X)n f?g???? ??????? ?????g??????? ?? ????? ?? ????? ????

8x2C x= supfg(y)jy2C??y < xg:

c= supfg(y)jy2C??y < cg= supfg(y)jy2Ig= supfg(y)jy2D??y < dg=d =2I n??? ?? ??????? ??? ??? ??a2K? ?????C???? ?????x =2C? ????y?? ??????? ??x???C???? ???? ???? ??? ?? ??????

8z2Chzy;xyi 0

hxy;xyi=2>0 fz2Hj hz;ni=ag????n=xy??a=hy;ni+22

8x;y2E82[0;1]p(x+ (1)y)p(x) + (1)p(y):

8x2M8t2Rh(x+tv) :=g(x) +t;

8x2M8t2Rg(x) +tp(x+tv):

8x2M8t >0g(x) +tp(x+tv)??g(x)tp(xtv);

8x2M8t >0ax;tbx;t????ax;t:=g(x)p(xtv)t

??bx;t:=p(x+tv)g(x)t sup x2M;t>0ax;tinfx2M;t>0bx;t

8x;y2M8s;t >0ax;sby;t:

??g? g(x)p(xsv)s p(y+tv)g(y)t car tg(x) +sg(y)tp(xsv) +sp(y+tv) car tp(xsv) +sp(y+tv)s+tpt(xsv) +s(y+tv)s+t =ptx+sys+t gtx+sys+t =tg(x) +sg(y)s+t: r kyf0k r x:=yf0kyf0k;

8f2Fkxfk=ky(f0+kyf0kf)kkyf0kkyf0kr:

?? ???? ????y2`1?? ????x2c0? ?? ????y(x) :=P1 n=0ynxn? ??????? ???y??? ??? ????? n=0xnen???? ????x:= (xn)n2c0? kyk1 jjjjjj? ??????? ???= y? jjjTjjj:= sup x2BEkT(x)kF: ?? ??????T2 Lc(E;F)??K2 K(E;F)???? ???jjjTKjjj<"2 ? ??????? ???T(BE)???? ???? ??? ??????? ???K(E;F)??? ????? ????Lc(E;F)? ???? ????x2BE? ???Pn(H) =Mn??jjjKPnKjjj 1n X ny nxn X njynjjxnj X njynjsup mjxmj=kyk1kxk1: ?? ???? ????x2c0? ?? ?????xPN ????? ??? ??? ???? ????? ?supn>Njxnj? ??? ???? ????0?????N! 1? ?? ???? ???? ??????n? ????un2C??? ???jynj=unyn??junj= 1???yn= 0? ?? ???? ??????? ??? ???????un= 1?? ??????PN n=0jynj=PN n=0unen PN n=0unen

1= supnNjunj= 1?

????PN n=0jynj jjjjjj? ?? ???? ???? ????N2N? ?? ??? ?????? ???P1 ???y2`1??kyk1 jjjjjj? ???? ????x2c0?? ????N2N?P nNxnen=P nNxnyn= yP (x) = y(x)? ?? ? ????= y? ?????T1+T2????? ??? ?? ??????BE????T

1(BE) +T

2(BE)??? ??? ??????? ??????? ????F

? ??T(BE) ? ????Mn??quotesdbs_dbs17.pdfusesText_23

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