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# h6color - ECE 580 Math 587 FALL 2009 Correspondence 16...

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ECE 580 / Math 587 FALL 2009 Correspondence # 16 April 12, 2011 ASSIGNMENT 6 Reading Assignment: Text: Chapter 5 (sects 5.10-5.13) and Chapter 6 Recommended Reading: Curtain & Pritchard: pp. 65-67, and Chapter 12; Balakrishnan: pp. 62-80, and Chapter 2. Advance Reading: Text: Chapter 7 Problems (to be handed in): Due Date: Thursday, April 21. 41. Let X be a Hilbert space, and { x n } be a sequence in X , converging weakly to x o X . i) Show that the convergence is also in the strong sense (that is in norm) if further the sequence of real numbers {∥ x n ∥} converges to the norm of x o , x o . ii) Again for the original problem, show that one can find a subsequence { x n k } such that the sequence of arithmetic means y m = 1 m m k =1 x n k , m = 1 , 2 , . . . converges strongly to x 0 (that is, y m x o ∥ → 0). 42. Let X be a real normed linear space, and X * be its dual. i) Show that a linear functional f on X is weakly continuous if and only if it is of the form f ( x ) = < x, x * > , for some x * X * .

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h6color - ECE 580 Math 587 FALL 2009 Correspondence 16...

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