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1. (i) Let (E,k · k) be a normed vector space. Consider a sequence (x(k))1k =1 in E. What does it mean that (x(k))1k =1 is a Cauchy sequence? What does it mean that (E,k · k) is a Banach space? [3 Marks] (ii) Let (E,k · k) be a Banach space. For n 2 N, n # 2, consider the vector space En . (a) Consider the norm k · k1 on En given by kxk1 = Xn l=1 kxlk for x = (x1, x2, . . . ,xn) 2 En. Prove that (En,k · k1) is a Banach space. (You do not need to prove that k·k1 is indeed a norm on En .) [6 Marks] (b) Now consider the norm k · k1 on En given by kxk1= max l2{1,...,n} kxlk for x = (x1, x2, . . . ,xn) 2 En. Prove that the norms k · k1 and k · k1 Total marks 12 are equivalent. [3 Marks] 2. (i) Let H be a Hilbert space and let G be a closed subspace of H. State the theorem on the orthogonal projection of a vector x 2 H onto G. [2 Marks] (ii) In the Hilbert space `2(R) , consider the vectors f = (1, 1, 1, 0, 0, 0, . . . ), g= (1, 0, 2, 0, 0, 0, . . . ). Consider the closed subspace G of `2(R) given by G = {af + bg | a, b 2 R}. (a) Find G?. [4 Marks] (b) Let P1 and P2 denote the orthogonal projections of the Hilbert space `2(R) onto the closed subspaces G and G? , respectively. Let x = (0, 1, 0, x4, x5, x6, . . . ) 2 `2(R), where xn = 2−n for n # 4. Find P1x and P2x. I 3. (i) Let E be a Banach space with norm k · k. Let A be a bounded linear operator on E, i.e., A 2 L(E) . Give two equivalent definitions of the norm of A. Let L(E) denote the space of all bounded linear operators on E, equipped with the operator norm k · k. What can you say about the space (L(E),k · k)? (You do not need to justify/prove your statement.) [3 Marks] (ii) Let p 2 [1,1) . Consider the following linear operator on `p(R) : Ax = (0, 0, x3,−x2, x5,−x4, x7,−x6, . . . ), x= (x1, x2, x3, . . . ) 2 `p(R). Prove that A 2 L(`p(R)) and find kAk. [7 Marks] 4. Let H be a Hilbert space and let A 2 L(H) . (i) Give the definition of the adjoint operator A⇤ . [2 Marks] (ii) State the theorem about the kernel and range of A⇤ . Moreover, prove the part of this theorem that concerns the kernel of A⇤. [6 Marks] (iii) Let now H = L2([1,1) ! C) and let A 2 L(L2([1,1) ! C)) be the bounded linear operator defined by (Af)(x) = (2+i)f(x3), x2 [1,1). Find A⇤ . 5. (i) Let A be a bounded linear operator on a Hilbert space H. What does it mean for the operator A to be unitary? Give both equivalent definitions! [3 Marks] (ii) State the theorem about the inverse of a unitary operator. (iii) Give an example of a unitary operator on the Hilbert space `2(C) (which is not a multiple of the identity operator), and explain why this operator is indeed unitary. [8 Marks] 6. (i) Let E be a Banach space over C, and let A 2 L(E) . (a) State the definition of both the set of regular points of A, and the spectrum of A. [3 Marks] (b) Let z be a regular point of the operator A. What is the resolvent of A at point z? State the Hilbert identity. [3Marks] (ii) Define A 2 L(L2([0, 2] ! C)) by (Af)(x) = ↵(x)f(x), x2 [0, 2], f 2 L2([0, 2] ! C). Here ↵(x) = 1X n=1 (−1)n 2 n #( 2 n+1 , 2 n ](x), x2 [0, 2], where #( 2 n+1 , 2 n ](·) denotes the indicator function of the set " 2 n+1 , 2 n ⇤ . Find the spectrum of A. [11 Marks] 1. (a) Let E1 and E2 be normed vector spaces. Let A : E1 ! E2 be a linear operator. Explain what it means that the operator A is bounded. State two equivalent definitions of the norm of A. [3 Marks] (b) Let p 2 [1,1) . Consider the following operator on the space E = lp(C) : for any x = (x1, x2, x3, . . .) 2 E, Ax = ⇣ 0, 2x3,− i 4 x4, x5,− i 6 x6, x7,− i 8 x8, . . . ,− i 2k x2k, x2k+1, . . . ⌘ . Prove that A 2 L(E) and find kAk. [8 Marks] 2. (a) Give two equivalent definitions of an invertible operator. Prove their equivalence. [4 Marks] (b) Formulate Banach’s theorem on inverse operators. [1 Mark] (c) Let p 2 [1,1) and E = Lp(R) := Lp(R ! R) . Let the operator A be defined by (Af)(x) = 4 2 + x4 f(x), x2 R, f 2 E. Prove that A 2 L(E) and kAk  2. [8 Marks] (d) Prove that the operator A defined in (c) is not invertible. [12 Marks] Hint: Find g 2 E of the form g(x) = 4 (2 + x4)↵, x2 R, where ↵ 2 R, such that g does not belong to Ran(A) . II 3. (a) Let H be a Hilbert space over C. Let A 2 L(H) . Give the definition of the adjoint operator A⇤ of A. [3 Marks] (b) Let H = L2([1,1) ! C) . Let A 2 L(H) be given by (Af)(x) = (2+i) f(3x2 − 2), x2 R, f 2 H. Find A⇤ . [12 Marks] 4. (a) Give the definition of a Hilbert–Schmidt operator in a Hilbert space. Prove that this definition is independent on the choice of a basis. [4 Marks] (b) Let H = l2(R) and consider A 2 L(H) given by A(x1, x2, x3, x4, x5, x6, . . .) = (0, 0, x2, x1, x4, x3, x6, x5, . . .), x2 H. Prove that A is an isometry, but not a unitary operator. [8 Marks] 5. (a) Let E be a Banach space over C, and let A 2 L(E) . State the definition of both the set of regular points of A, and the spectrum of A. [3 Marks] (b) Define A 2 L(L2([0, 2] ! C)) by (Af)(x) = ↵(x)f(x), x2 [0, 2], f 2 L2([0, 2] ! C). Here ↵(x) = 1X n=1(−1)n 2 n ↵n(x), x2 [0, 2], where ↵n denotes the indicator function of the set ✓ 2 n + 1 , 2 n % , i.e. ↵n(x) = 8< : 1, x2 ✓ 2 n + 1 , 2 n % , 0, otherwise. Find the spectrum of A.

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