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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) e-x , 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 A-1 is linear. (3) b) Is A-1 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