[PDF] topologie-des-espaces-normés.pdf

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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??? ??? uupquotesdbs_dbs6.pdfusesText_12

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