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# h5color - ECE 580 Math 587 SPRING 2011 Correspondence 14...

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ECE 580 / Math 587 SPRING 2011 Correspondence # 14 March 30, 2011 ASSIGNMENT 5 Reading Assignment: Text: Chapter 5, Sections 1-9 Advance Reading: Text: Chapter 5, Sections 10-13. Recommended Reading: Curtain & Pritchard: pp. 65-67, and Chapter 12; Balakrishnan: pp. 62-80, and Chapter 2. Problems (to be handed in): Due Date: Tuesday, April 12 . 34. Let X = C [0 , 1] with the standard maximum norm, and f X * be defined by f ( x ) = x (1 / 4) + 3 x (1 / 2) + 2 1 0 tx ( t ) dt i) Compute f , the norm of f . ii) Obtain a function of bounded variation on [0 , 1], say v ( · ), such that v (0) = 0 and f ( x ) = 1 0 x ( t ) dv ( t ) 35. Let g 1 , g 2 , . . . , g n be linearly independent linear functionals on a vector space X . Let f be another linear functional on X such that for every x X satisfying g i ( x ) = 0 , i = 1 , 2 , . . . , n , we have f ( x ) = 0. Show that there exist constants λ 1 , λ 2 , . . . , λ n such that f = n i =1 λ i g i . Hint : Use the Hahn-Banach Theorem (extension form; Correspondence 13). 36. Let c 0 be the space of all infinite sequences of real numbers converging to zero , endowed with the max norm. Let x = { ξ i }

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h5color - ECE 580 Math 587 SPRING 2011 Correspondence 14...

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