Functional Analysis Problems

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 dierentiable 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 dierentiable 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 dierentiable 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 dened by F(x) = jjxjj : (b) F : Rnnf0g ! R dened by F(x) = jjxjj x, where 2 R is a given number; (c) F : Rn ! Rn dened 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 identied with R4, i.e. every matrix X 2 L(R2); X =  x1 x2 x3 x4  is identied with the vector [x1; x2; x3; x4]. We dene the map F : L(R2 ! L(R2)); F(X) = X2; Compute the Jacobi matrix of F at A =  a b c d  2

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