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p i(x1;x2) =xi? ???? O?? ?????? ??R2? ??????? ???p1(O)??p2(O)???? ??? ??????? ??R? ???? H=(x;y)2R2xy= 1? ??????? ???H??? ?? ????? ??R2?? ??? p
1(H)??p2(H)?? ???? ??? ??? ?????? ??R?
??? ??????? ?? F??? ????? ?? ???p2(F)??? ?????? ?????p1(F)??? ?????? F=E? ??????F??? ?????? ?????? ??? ???? ???? ?????? ?????E??x2E? ??????? d(x;F) = 0()x2F? ????? ?F??? ?????? ?????? ??? ???? ??E??x2E? ??????? d(x;F) = 0()x2F? ??????? ????? ?????? ???? ???????U??V???? ??? d(A;B) = infx2A;y2Bd(x;y)>0? kuk1= sup n2Njunj? ????A??? ?????? ??? ???? ??R????? ??? ???? ????x???? ?? ?????? ?? ?? ?? ???? kuk1= supn(junj)? ??? ??????? ??????? ?????? ?????? (an)??? ???? ????? ??????? ????? ????? kak=+1X n=0janj? ??? ??????? k k??? ??? ????? ???E? F=( a2E+1X n=0a n= 1) ??????E=C([0;1];R)????? ???k k1?? ?? ?????? A= f2Ef(0) = 0??Z 1 0 f(t)dt1 ??? ??????? A??? ??? ?????? ???????
8f2A;kfk1>1?
A=(x;y)2R2xy= 1??B=f0g R
??????? ???Z??? ??? ?????? ?????? ??R? N
1(P) = sup
t2[0;1]
P(t)??N2(P) = sup
t2[1;2] P(t)? =P2EP(0)6= 0 ?????? ?????? ???? ?? ?????N1? ???? ?? ?????N2?
Vect(A)VectA?
Fr(Fr(F)) = Fr(F)?
??? ??????? A\B=;=)A\B=;? d(A;B) =d(A;B) ??? ??????d(A;B) = infx2A;y2Bd(x;y)? ??????? S n i=1Ai=Sn i=1A i? n i=1Ai??Tn i=1A i? kfk1;A=kfk1;A ????A??? ?????? ???? ?????? ?????E?
U=M2 M2(C)82SpM;jj= 1??
R=M2 M2(C)9n2N;Mn=I2?
??? ??????? U??? ??? ?????? ?????? ??M2(C)? ??????? ???Rp??? ?? ?????? ??Mn(K)? u(x) = 1??? ??????? ff=f? x2[0;1]f(x) =x ff=f? 8x2E; u(x) kxk? ???? ????n2N? ?? ???? v n=1n+ 1n X k=0u k?
Im(uId)\Ker(uId) =f0g?
Im(uId)Ker(uId) =E?
Im(uId)Ker(uId) =E
?????? ??E? ??y2R;f1(fyg)??? ??? ?????? ??????? ??????? ???GLn(R)??? ????? ????Mn(R)? ??????? ??? U\V??? ?????? ?? ?????? ????? ??E? ??????(un)n2N??(vn)n2N???? ?????? ??????? ?????? ??? u n!+1;vn!+1??un+1un!0? ????? ?" >0??n02N??? ??? ???? ????nn0?jun+1unj "? ??????? ??? ???? ????aun0? ?? ??????nn0??? ???junaj "? ?? ??????? ??? funvpjn;p2Ng??? ????? ????R? ??????? ???mlnn(m;n)2ZN??? ????? ????R? ?? ???? ???a >0? ???????a2H????H=aZ? ?? ???? ???a= 0? ??????? ???H??? ????? ????R? ??? ??????? cos(n)n2N??? ????? ????[1;1]? ??? ??????? cos(lnn)n2N??? ????? ????[1;1]? M n(C)? kuk1=+1X n=0junj? kuk1= sup n2Njunj??
2:f7!sZ
+1 0 f2(t)dt? ??????? ???A=E?
8a;b2A;a+b2
2A?
8(a;b)2A2;pab2A?
??????? ???A\(RnQ)??? ????? ????]infA;supA[? N '(f) =kf'k1? [0;1]? n!1? A=umu n m > n ??????A??B???? ??????? ?????? ???? ?????? ?????E?
8x;y2R;f(x+y) =f(x) +f(y)?
8(x;y)2R2; fx+y2
=12 f(x) +f(y)? ??? ??????? D=p=2np2Z;n2N??? ????? ????R? ??????? ??? ???? ????A;B2 Mn(C)?AB=BA? ????n2N????n2? ????? ?A2 Mn(C)??P2GLn(C)? ??A? eA? eeA? ????(un)??? ????? ?????? ????? ???un+1un!0??un!+1? ????(vp)??? ????? ?????? ????? ???vp!+1? ?? ??? ???? ????? a??b???? ???a < b? ????p??q????N? ?? ???? (wn) = (un+pvq)? ??????? ??? ???? ???? ???????p??q?? ????? ????? ??? ???? ???w0a??? ????n2N?jwn+1wnj (ba)=2? ??? ??????? unvp(n;p)2N2??? ????? ????R? af(t)dt= 0? ??????? ????? ?????? ??? ????? Z b a P n(t)dt= 0??sup t2[a;b] f(t)Pn(t)!n!+10? N
1(fPn)!0??N1(f0P0n)!0?
k2 f0;:::;ng? ?? ???? B n;k(x) =n k x k(1x)nk? nX k=0B n;k(x);nX k=0kB n;k(x)??nX k=0k
2Bn;k(x)?
????? ? >0??x2[0;1]? ?? ?????
A=k2J0;nKjk=nxj ??B=k2J0;nKjk=nxj< ?
X k2AB n;k(x)14n2? f n(x) =nX k=0fkn B n;k(x)? a n=Z 1
1(1t2)ndt
n(x) =1a n(1x2)n? R1
0t(1t2)ndt? ?? ??????? ???
a n=Z 1
1(1t2)ndt1n+ 1?
????? ???[;1]? f n(x) =Z 1
1f(xt)'n(t)dt
f(x)fn(x) =Z 1
1(f(x)f(xt))'n(t)dt?
????f2 C([a;b];R)? ?? ??????? ??? ???? ????n2N? Z b a xnf(x)dx= 0? I n=Z +1 0 xne(1i)xdx? ?? ??????? ????? ?????? f????C([0;+1[;R)??? ?????? ????? ???? ???? ????n ????N? ?? ???Z+1 0 xnf(x)dx= 0? lim n!+1 Zb a f(x)sin(nx)dx =2 Z b a f(x)dx? ??????F?? ????? ?? ???? ????n2N? O n=[ a2FB(a;1=n) O a ???F??? ?????? ???? x2p1(O)? ?? ??????y2R??? ???a= (x;y)2O? ?????O??? ??????? ?? ??????" >0??? ???B1(a;")O?? ?????]x";x+"[p1(O)? ?????p1(O) ?? ?? ????p2(O)??? ??????? ???? ((xn;yn))n2N2HN????? ???(xn;yn)!(x;y)? ?????xnyn= 1? ? ?? ??????xy= 1? p
2(H) =R?? ???? ??? ?????? ????R?
???? (xn)n2N2(p1(F))N????? ???xn!x? ????n2N? ?? ??????yn??? ??? (xn;yn)2F? ??????y= limy'(n)? ?????F??? ??????(x;y) = lim(x'(n);y'(n))2F???? x=p1((x;y))2p1(F)? 0
E2F???? ?? ?????? >0??? ???B(0E;)F?
???? ????x2E? ?? ???? ?????? x=y ????y2B(0E;)?????? ??????? d(x;F) = infkxyky2F ((= )??x2F?????02kxyky2F?? ????d(x;F) = 0 kxynk 1n+ 1? d(x;F) = infkxyky2F ((= )??x2F?????02kxyky2F?? ????d(x;F) = 0 kxynk 1n+ 1? U=[ x2FB x;12 d(x;G) ??V=[ x2GB x;12 d(x;F) ???? ??????y2U\V????? ?? ??????a2F??b2G???? ??? d(a;y)<12 d(a;G)??d(b;y)<12 d(b;F)? d(a;G);d(b;F)d(a;b) d(a;b)d(a;y) +d(y;b)< d(a;b)?
U\V=;?
U=[ a2AB(a;d=2)??V=[ b2BB(b;d=2) U\V6=;=) 9(a;b)2AB;B(a;d=2)\B(b;d=2)6=;=)d(A;B)< d? ?????u= (un)???? ?? ?????k k1????? ???? ????n2N?? ????p2N? u pnup n+1??? ????? ? ?? ??????unun+1?? ????u2A? ?????u= (un)???? ?? ?????k k1????? ???? ????" >0?? ??????p2N??? ??? uup
1"=2?? ???????upn!n!+10? ?? ??????N2N??? ???
8nN;upn"=2
junj unupn+upn"? ?????u!0?? ????u2B? ??? ?????u= (un)???? ?? ?????k k1????? ?? ??????`p?? ?????? ??up? ?? ????? (`p)??? ??? ????? ?? ?????? ???????`p`q upuq
1? ??????`?? ?????? ??
u2C? ?????u= (un)???? ?? ?????k k1????? ???? ????" >0?? ??????p2N??? ??? uup ??n???? ???upn"=2?? ???? ???? ??? junj unupn+upn"? u p=pX k=112 up+qup 1p+qX k=p+112 k12 p!0 u
0= limp!+1p
X k=112 k= 1 ?? ???un<1???? ????n??????? ???? ???un= 1?? ????kjn???? ????k2N? ??? ?????? ???? R(N)???? ?????? ????R(N) B(N;R)? ?? R(N)??? ?????? ????? ???????02R(N)?? ?????? >0??? ??? B
1(0;)R(N)?
?????? ???? ??????? ???? ????B1(0;)6R(N)?? ????R(N)????? ??? ??????? ? ???N2N? ??????uN?????? ???uNn=1n+1??nN??uNn= 0?????? (uN)2R(N)??uN!u????u????? ???un=1n+1? ?? ???? uNu
1=1N+ 2!0?
????u =2R(N)????R(N)????? ??? ?????? jxyj=d(x;A)? ??d(x;A) = 0????x=y2A? ?????A??? ?????? a;b2A??c =2A? ??????= supfx2Ajxcg??= inffx2Ajxcg? ?? ?;2A? < c < ??];[CRA? =+2 ? ?? ?d( ;A) =2 =j j=j ???? ??????n? ??????`n= limun= limp!+1unp? ????(un)?? ????? ?????? ???
8n2N;8p2N;unp=1(p+ 1)1+1=n?
??????? ??A? ?? ??????A????? ??? ??????? ka+bk=+1X n=0jan+bnj +1X n=0 janj+jbnj=kak+kbk k:ak=+1X n=0janj=+1X n=0jjjanj=jj+1X n=0janj=jjkak? ????? ??kak= 0?????
8n2N;janj kak= 0
?????(an)n0= (0)n0 ': (an)n07!+1X n=0a n??? ??????
8a= (an)n02E;'(a)=
+1X n=0a n +1X n=0janj=kak? ??????e= (1;0;0;:::)?? ?? ??????? ??F??
8 >0;e+e =2F??
e(e+e) ??????p=e+p:(1;1;0;0;:::)?
8p2N;p2F??
p !p!+1+1? ?? ??????F????? ???? ??? ??????? u =2F? ???? ????n2N? ?? ???? ??????xn=yn+nu????yn2F??n2K? ?????? ?????zn=1 nxn=1 nyn+u? ???????kxnk ! kxk??jnj !+1? ?? ?kznk !0?? ????1 nyn! u? 1 ????u2F?? ??? ?????? ?? ??????2K? x=y+u2F+ Vect(u)? ??????A1?? ?????? ??(Ap)? A ?? ????A=Ap???? ? ?? ??????A=A1? A=1 0 ????Pp=p0 0 1 P
1pAPp=1=p
0 !I2
Ker(AI2)26= Ker(AI2)?
????X22Ker(AI2)2nKer(AI2)??X1= (AI2)X2? ?? ???????(X1;X2) T=0 @1 () 0() (0) (0)B1 A ????Pp= diag(p;1;:::;1)? ?? ??????? P
1pTPp=0
@1=p(=p) 0() (0) (0)B1 A !0 @0 (0) 0() (0) (0)B1 A =A1? ???A2 Mn(R) P ?? ??????t2R??? ???P0=Q+tR2GLn(R)?? ???? ????? ???????AP0=P0A1?
Ker(AI2)26= Ker(AI2)?
????X22Ker(AI2)2nKer(AI2)??X1= (AI2)X2? ?? ???????(X1;X2) ??????AX1=X1??AX2=X2+X1? ??2CnR? ?? ???? ??????=a+ ib????b2R? ??????X3=X
1??X4=X
2? ?? ???????(X1;X2;X3;X4)??? ????? ???6=? ???????VectC(Y1;:::;Y4) = VectC(X1;:::;X4)? ?? ???????(Y1;:::;Y4)??? ????? ?? ?? ?????? ??? ?? ??????AY1=aY1bY3?AY2=aY2bY4+Y1?AY3=aY3+bY1 M n(R)? ?? ???????T O B T=0 B
B@a1b0
0a0b b0a1 0b0a1 C CA? ????Pp= diag(p;1;p;1;:::1)? ?? ??????? P
1pTPp!T10
O B =A1 T 1=0 B
B@a0b0
0a0b b0a0 0b0a1 C CA? ????Mn(R)? f n(0)!f1(0)? ?? ?? ??????f1(0) = 0? ?????Z1 0 f n(t)dt!Z 1 0 f
1(t)dt
Z1 0 f
1(t)dt1?
Z 1 0 f(t)dtZ 1 0 f(t)dtZ 1 0 kfk1dt1 Z1 0 f(t)dt= 1 Z1
01f(t)dt= 0?
???? ????n2N? ?? ???? ??????un= (xn;yn)????xnyn= 1? ? ?? ?????? ?? ???????x1y1= 1?? ????u1= 1? ?????? ???A??? ??????? u n= (1=n;0) = (1=n;n) + (0;n)2A+B? ?????n!+1?un!(0;0)? RnZ=[ n2Z]n;n+ 1[? ????"= 1=2? ?? ?????? ?? ????N2N??? ???
8nN;jxn`j<1=2
8m;nN;jxmxnj<1?
8m;nN;xm=xn?
Z=f1(f0g)
????f0g?????? ?????? ??R? ='1R??? ?? ?????? ??????? ?E?????? ?????? ??E???? ?? ?????N1? sup t2[0;2]
Pn(t)f(t)!0
P n(0)!0??N2(PnP)!0? Q n=PnPn(0)? ???? ????n2N?Qn(0) = 0????Qn=2 N
2(Qn)N2(PnP) +Pn(0)!0
Q nN 2!P? ???????AVectA? ?? ?AVectA?
Vect(A)VectA?
Fr(A) =AnA=A\{EA=A\{
EA?
Fr(F) =F\{
EF
Fr(Fr(F)) = Fr(F)\{
EFr(F)?
??Fr(F)F=F????{EF{EFr(F)????{ EF{ EFrF? ?? ????FrF{
EF????FrF{
EFrF????
quotesdbs_dbs43.pdfusesText_43
Topologie des espaces normés - Xiffr