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Unformatted text preview: 1. a) Give the definition of a contraction mapping. (2) b) Prove that any contraction mapping is continuous. (2) c) Is it true that any continuous mapping of a Banach space to itself is a contraction? If "yes", prove it, if "no", give an example. (2) 2. 1 a) Prove that the algebraic equation x = (1 + x) 5 has a root inside the segment [1, 2]. (3) b) Find any three successive approximations to this root (only algebraic expressions, without evaluation). (2) 3. A function f (x) : [0, 1] R is measurable with respect to the usual  additive measure defined on the  algebra of all measurable subsets of [0, 1]. Prove that ef (x) is also measurable. (3) 4. Evaluate the Lebesque integral
R+ f (x) d, where f (x) = Explain your answer. (3) ex , x R+ \ N; ex , x N. 5. Given f (x) = evaluate a) f (x) L (0,1) , (2) b) f (x) L2 (0,1) . (2) Explain your answer. 1, x [0, 1] Q; 2, otherwise, 6. Consider the operator
t T (x(t)) =
0 x( ) d from C[0, 2] to C[0, 2], Prove that the operator is a) linear (2) b) bounded (2) c) continuous. (1) d) Evaluate its norm. (2) x(t) C[0,2] = sup x(t).
0t2 7. Let A : X Y be a linear bounded invertible operator, where both X and Y are Banach spaces. a) Prove that the inverse operator A1 is linear. (3) b) Is A1 necessarily bounded? Explain your answer. (3) 8. a) Give the definition of the dual space of a normed linear space X. (2) b) Describe the space l3 . (2) c) What space is dual to l3 ? (Only answer, without proof). (2) ...
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This note was uploaded on 09/21/2009 for the course CS 580 taught by Professor Fdfdf during the Spring '09 term at University of Toronto.
 Spring '09
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