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Unformatted text preview: Modern Analysis, Homework 9 Answers available on Aug 10, or sooner Review problems for chapter 9. Recommended problems are marked with a *. Through out, α is an increasing function on the interval [ a,b ]. 1. * [10] Rudin, Ch 6, #1: Suppose that a ≤ x ≤ b , α is continuous at x , f ( x ) = 1, and f ( x ) = 0 if x 6 = x . Prove that f ∈ R ( α, [ a,b ]) and that R b a f dα = 0. 2. * [14] Rudin, Ch 6, #2: Suppose that f ≥ 0, f is continuous on [ a,b ] and that R b a f ( x ) dx = 0. Prove that f ( x ) = 0 for all x ∈ [ a,b ]. 3. [15] Rudin, Ch 6, #11: For u ∈ R ( α, [ a,b ]) define k u k 2 = Z b a  u  2 dα 1 / 2 . Given f,g,h ∈ R ( α, [ a,b ]), prove that k f h k 2 ≤ k f g k 2 + k g h k 2 . 4. [28] Rudin, Ch 6, #12: With the notations of the previous exercise, given ε > 0 and f ∈ R ( α, [ a,b ]), prove that there exists a continuous function g on [ a,b ] such that k f g k 2 < ε ....
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This note was uploaded on 10/18/2011 for the course MATH S4061X taught by Professor Peters during the Spring '11 term at Columbia.
 Spring '11
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