081023FunctAnalysis - 1 a Give the definition of a...

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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. a) Prove that the algebraic equation x = (1 + x ) 1 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 σ - additive measure defined on the σ - algebra of all measurable subsets of [0 , 1] . Prove that e f ( x ) is also measurable. (3) 4. Evaluate the Lebesque integral Z R + f ( x ) dμ, where f ( x ) = e - x , x R + \ N ; e x , x N . Explain your answer. (3) 5. Given f ( x ) = 1 , x [0 , 1] Q ; - 2 , otherwise , evaluate a) k f ( x ) k L
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