+1 862 207 3288
Instructions. Please solve the following problems (show all your work). You can use your textbooks,
class notes. Work on your own and do not discuss the problems with your classmates or anyone else.
Please let me know if you have any questions concerning the problems or you do need some hints.
1. (20 pts) Find the norm of the following linear operator
A : R3 ! R2;
A(x; y; z) = (x y; x z) :
2. (20 pts) Let K : [0;1] [0;1] ! R be continuous. DeÖne
A : C ([0;1]) ! C ([0;1]) by
A(f)(t) = Z01 K (t; s) f (s) ds:
Show that A 2 L(C ([0;1]) ; C ([0;1])) and
kAk sup Z01 jK (t; s)j ds j t 2 [0;1] :
3. (20 pts) Find the norm of the operator
T : C ([0; 1] ; R) ! C ([0; 1] ; R) ;
(Tx)(t) = Z01 cos (t s) x(s) ds; x 2 C ([0; 1]) ; t 2 [0; 1] :
4. (20 pts) Ler l2 denote the Hilbert space of all real sequences fxng such that
1 X i
=1
jxij2 is convergent,
that is
l2 = (fxng j X i1 =1 jxij2 < 1) :
Recall, h ; i : l2 ! R; be the inner product on l2,
hfxng ; fyngi =
1 X i
=1
xiyi:
DeÖne S = fei j i 2 Ng ; where ei = (0;0; :::;1;0; :::) (1 on ith position), i = 1;2; ::: : Show that S
is a complete orthonormal set in l2.
5. (20 pts) Let E be a Banach space. Denote by L(E) the Banach space of all bounded linear operators
T : E ! E (equipped with the operator norm
kTk = sup fkT (x)k j kxk 1; x 2 Eg 🙂
Assume that F : L(E) ! L(E) is given by the formula:
1
a) F(X) = XTX2
b) F(X) = (X + T)2
c) F(X) = TX2 + XTX + X2T
where X 2 L(E) and T : E ! E is a Öxed bounded linear operator.
Compute the derivative of F at X 2 L(E), i.e. DF(X) : L(E) ! L(E).
6. (40 pts, Extra Credit) Show that the following linear mapping
A : L2 ([0;1]) ! L2 ([0;1]) ; given by
A(f)(t) = Z01 tsf (s) ds
is continuous. Find its norm.
2
TAKE ADVANTAGE OF OUR PROMOTIONAL DISCOUNT DISPLAYED ON THE WEBSITE AND GET A DISCOUNT FOR YOUR PAPER NOW!
