# exam1 - Intro to Modern Analysis II Math S4061X Section 2...

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Unformatted text preview: Intro to Modern Analysis II, Math S4061X Section 2 Summer 2011 Exam 1 Name: July 25, 2011 Do all problems, in any order. No notes, texts, or calculators may be used on this exam. Problem Possible Points Points Earned 1 10 2 10 3 10 4 5 5 3 TOTAL 38 1. Let { f n } n ∈ N be a sequence of continuous functions defined on a compact space ( K,d ). Suppose that there exists a continuous function f on K and that for every convergent sequence of points { x n } n ∈ N (say x n → x ), we have lim n →∞ f n ( x n ) = f ( x ) . (a) (7 points) Prove that f n → f uniformly on K . (b) (3 points) Show that the conclusion in (a) is false if we do no assume that K is compact. 2. (a) (5 points) Suppose that f is a 2 π-periodic function on R , Riemann-integrable on [- π,π ]. Let ˆ f ( n ) denote the n-th Fourier coefficient of f , ie ˆ f ( n ) = 1 2 π Z π- π f ( x ) e- inx dx, and suppose that ˆ f ( n ) = 0 for all n ∈ Z . Prove that f ( x ) = 0 whenever f is continuous at x ....
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exam1 - Intro to Modern Analysis II Math S4061X Section 2...

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