TL THI DAI HOC MON GIAI TICH HAM.doc
Transcript of TL THI DAI HOC MON GIAI TICH HAM.doc
BI TP CHNG I
BI TP CHNG I
1.5/ Gi s E=C[0,1] v l mt hm c nh, A:E ( E l mt nh x xc nh bi (A(x))(t) = Chng minh rng A tuyn tnh v lin tc. Xc nh chun ca A.Gii
+ Chng minh A tuyn tnh:Ta cn chng minh
:
=
=
=
=
Vy
( A tuyn tnh+ Chng minh A lin tc:
(
( A b chn
Kt hp vi A tuyn tnh ta c A lin tc
+ Tnh
T chng minh b chn ta c
(1)
Chn ,
(
(
(2)
T(1), (2) suy ra :
1.6/ Gi s X= v nh x c xc nh bi
a/ Chng t X l khng gian con Banach ca C[0,1]
b/ CMR: l nh x tuyn tnh lin tc. Tm
c/ Hi c l n nh, ton nh khng?
Gii
a/(
( khi
Ta c:
(
(1)
khi
Ta c:
(
(2)
T (1) v (2) suy ra X l khng gian con(*)
Vi C[0,1] l khng gian Banach nn chng minh X l khng gian Banach ta chng minh X ng trong [0,1]
V nn
Vy ( X l khng gian Banach con
b/ + Chng minh tuyn tnh:
Ta cn chng minh
:
=
=
=
=
=
Vy
( tuyn tnh
+ Chng minh lin tc:
=
( ( b chnKt hp vi tuyn tnh ta c lin tc
+ Tnh
T chng minh b chn ta c (1)
Chn
(
(
Cho , ta c
(2)
T(1), (2) suy ra :
c/ + n nh khng?
( n nh
+ ton nh khng?
Gi s
Vy
( x khng lin tc (v l v )
( khng l ton nh1.7/ Gi s X, Y l hai kgc. CMR
a/
b/ S hi t trong X x Y tng ng vi s hi t theo ta , ngha l trong X x Y trong X v trong Y
c/ l dy Cauchy trong X x Y ln lt l dy Cauchy trong X v YGii
a/ +
+
=
+
=
=
b/
c/ Cauchy trong X x Y
1.10/ Cho l mt khng gian nh chun trn K. Tm iu kin ca cng thc , cng l chun trn EGii
*
Do tha mn yu cu
*
khng l chun khng tha mn
+ l chun trn ETht vy:
2.8/
Bit rng khvt vi hai php ton thng thng ca dy l khng gian Banach vi chun . Xt nh x xc nh bi cng thc:
a/ CMR nh x f l tuyn tnh lin tc
b/ Tm
c/ Chng minh rng f l nh x m
Gii
a/
+ Chng minh f tuyn tnh:
Ta cn chng minh
=
=
=
=
Vy
( f tuyn tnh
+ Chng minh A lin tc:
(
( f b chn
Kt hp vi f tuyn tnh ta c f lin tc
+ Tnh
Vi M =
Vy
(1)
Cho
(
(
(2)T(1), (2) suy ra:
c/ Chng minh rng f l nh x m
f l nh x tuyn tnh lin tc vy chng minh f l ax m th ta cn chng minh f l ton nh
( f l ton nh
Vy f l mt nh x m
Bi 3.8 Xt KG Banach vi chun v t
a/ CMR cng thc x.nh 1 tch v hng trn b/ CMR vi tch v hng trong cu a l 1 kg Hilbertc/ CMR vi l phm hm tuyn tnh, lin tc trn . Tnh theo cu a
a/ +Dng Hilbert:
+ Dng
+ Xc nh dng
Vy l mt tch v hng trn
b/ l khng gian tin Hilmite vi tch v hng
Ta chng minh y vi chun sinh bi tch v hng
Ta c:
Vy l kg HB
c/ Ta c vi
( f l mt phm hm tuyn tnh lin tc
V
Vy
3.4/ Gi s E1, E2 l hai khng gian tin Hilbert
a/ CMR E1x E2 l khng gian tin Hilbert vi tch v hng ,
b/ CMR nu E1, E2 l hai khng gian Hilbert th E1x E2 cng l khng gian Hilbert
Gii
a/
+Dng Hilbert:
+ Dng
+ Xc nh dng
b/ chun sinh bi tch v hng
Gi s l dy Cauchy
Ta chng minh
Hilbert
3.5/ Cho l mt gii trc chun trong kgHb E cn l dy s b chn. CMR
a/ Chui hi t vi mi
b/ nh x l tuyn tnh lin tc. Tnh chun
Gii
a/ Ta c
Vi
Vy
hi tb/
+ Chng minh f tuyn tnh:
Ta cn chng minh
=
Vy
( f tuyn tnh
+ Chng minh f lin tc:
(
( f b chn
Kt hp vi f tuyn tnh ta c f lin tc
+ Tnh
T chng minh f b chn ta c:
(1)
(2)T(1), (2) suy ra:
3.3 Xt K khng gian vect
Vi mi
t
a/ CMR hi t
b/ CMR cng thc
d/ CMR vi l mt dy ltt trong
e/ p dng cng thc tnh trc giao ha Gram-Schmidt xc nh mt c s trc chun ca khng gian con vi c xc nh trong cu trn
gii
a/ Ta c
Vy
hi t
b/
+Dng Hilbert:
+ Dng
+ Xc nh dng
Vy l mt tch v hng trn
d/ CMR ltt trong
Ta c
Vy ltt trong
e/
m
Vy l c s trc chun ca
Bi 3.6: Xt KG CMR
a/ Cng thc vi mi xc nh 1 tch v hng trn
b/ vi tch v hng trn l 1 KG Hilbert
c/ Cng thc x.nh mt phim hm t. tnh lin tc trn . Tm
gii
a/ +Dng Hilbert:
+ Dng
+ Xc nh dng
Vy l mt tch v hng trn
b/
Vy l kg HB vi tch v hng cho
c/ Ta c
l mt phm hm tuyn tnh lin tc
Tm
l mt phm hm tuyn tnh lin tc
Cu kim tra:
Trnh by s gii bi tp: Cho nh x , CMR l mt nh x tuyn tnh lin tc v tm . Minh ha chi tit vi nh x vi mi
Gii
- S gii bi tp
* Chng minh f tuyn tnh
Ta cn chng minh:
* chng minh f lin tc: Ta cn chng minh f b chn
f b chn kt hp vi f tuyn tnh suy ra f lin tc
* Tnh
T chng minh f b chn
(1)
Ta cn chng minh
(2)
T (1) v (2) suy ra:
- Minh ha chi tit
* Chng minh f tuyn tnh
tuyn tnh
* Chng minh f lin tc
b chn
Kt hp vi f tuyn tnh ta c f lin tc
* Tnh
T chng minh f b chn
(1)
Chn
(2)
T (1) v (2) suy ra:
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