Functional Analysis Problems

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