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takehome final

# takehome final - ECE 580 Math 587 Professor Tamer Baar s...

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ECE 580 / Math 587 SPRING 2011 Professor Tamer Ba¸ sar May 5, 2011 FINAL EXAM DUE DATE & TIME : Monday, May 9, 2011; 9:00 am. If I am not around, leave with my secretary next door to my office at CSL, or slide under my door. Answer the four questions below, starting each one on a separate sheet. Problem 1 Let C [ 1 , 1] denote the normed vector space of continuous functions on [ 1 , 1], equipped with the standard maximum norm. We wish to find a linear functional f on X = C [ 1 , 1], with minimum norm, that satisfies the side conditions: f ( x 1 ) = 1 , f ( x 2 ) = 2 where x 1 ( t ) = 2 t 2 t , x 2 ( t ) = t 2 + t + 1 i) Is this a feasible problem? Justify your answer. ii) If your answer to part (i) is in the affirmative, obtain the solution. iii) Repeat (i) and (ii) above with X = L 2 [ 1 , 1], instead of C [ 1 , 1], where L 2 [ 1 , 1] is the standard Hilbert space of square-integrable functions on [ 1 , 1]. iv) Would the result in (iii) be any different if X is the normed vector space of continuous functions on [ 1 , 1] with the L 2 norm? v) Let x * be the solution you obtained in part (ii) . Describe the set of all x C [ 1 , 1] that are orthogonal to x * , that is < x, x * > = 0.

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