## Transcribed Text

1. Let (V; jj jj) and (W; jj jj) be normed spaces. Show that the following jj jj1; jj jj2; jj jj1 :
V W ! R given by
(a) jj(v;w)jj1 = jjvjj + jjwjj; where (v;w) 2 V W
(b) jj(v;w)2 =
jjvjj2 + jjwjj2
1=2
, where (u;w) 2 V W
(c) jj(v;w)jj1 = maxfjjvjj; jjwjj j (v;w) 2 V Wg are norms on V W.
2. Verify that the function jj jj1 : B(X; V ) ! R, given by
jjfjj1 = supfjjf(x)jj j x 2 Xg
is a norm on B(X; V ):
3. Find the norm of the operator
T : C([0; 1]) ! ([0; 1]);
(Tx)(t) =
Z 1
0
cos (t s)x(s)ds; x 2 C([0; 1]); t 2 [0; 1]:
4. Let C1([0; 1]) denotes the space of dierentiable functions f : [0; 1] ! R, such that
f0[0; 1] ! R is continuous. Show that the linear operator
A : C1([0; 1]) ! C([0; 1]); given by
A(f)(x) = f0(x); f 2 C1([0; 1]); x 2 [0; 1]
is not bounded.
5. 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)
jjTjj = supfjjT(x)jj j jjxjj 1; x 2 Eg
Assume that F : L(E) ! L(E) is given by the formula:
(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 xed bounded linear operator.
Compute the derivative of F at X 2 L(E), i.e. DF(X) : L(E) ! L(E)
6. Let E and F be two Banach spaces, U E an open set and F : U ! F; f : U ! R
two dierentiable at x0 2 U maps. Show that the product map
f F : U ! F; (f F)(x) = f(x) F(x); x 2 U;
is dierentiable at x0 and we have the following Product Formula:
D(f F)(x0)h = (Df(x0)h) F(x0) + f(x0) DF(x0)h; 8h 2 E:
7. Consider the Euclidean space Rn equipped with the norm
jjxjj =
vuut
Xn
i=1
x2i
1
Given the following maps
(a) F : Rnnf0g ! R dened by F(x) = jjxjj :
(b) F : Rnnf0g ! R dened by F(x) = jjxjjx, where 2 R is a given number;
(c) F : Rn ! Rn dened by
F(x) =
Ax p
1 + jjBxjj2
where A;B : Rn ! Rn are given two linear operators.
Use the known results (e.g. the Chain Rule, Product Rule, etc) to compute (i.e. to nd
an explicit formula) for the derivative DF(x).
8. Consider the space L(R2), which can be identied with R4, i.e. every matrix X 2
L(R2); X =
x1 x2
x3 x4
is identied with the vector [x1; x2; x3; x4]. We dene the map
F : L(R2 ! L(R2)); F(X) = X2;
Compute the Jacobi matrix of F at A =
a b
c d
2

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.