h2color - ECE 580 Math 587 SPRING 2011 Correspondence 4...

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ECE 580 / Math 587 SPRING 2011 Correspondence # 4 February 7, 2011 ASSIGNMENT 2 Reading Assignment: Text: Chp 10 (pp. 272-277); Correspondence # 2. Recommended Reading: Curtain & Pritchard: Chapters 1 (pp. 20-22), 4 (pp. 55-64); Balakrishnan: Chapter 1, and Chapter 2 (pp. 54-57). Liusternik & Sobolev: Chapter 1 (pp. 26-44). Advance Reading: Text: Chapter 3 Problems (to be handed in): Due Date: Thursday, February 17 . 11. Consider the space p consisting of all ordered numbers ( ξ 1 , ξ 2 , . . . , ξ k 1 ) where k 1 is a natural number and ξ i ’s are arbitrary real numbers. If x := ( ξ 1 , ξ 2 , . . . , ξ k 1 ) y := ( ν 1 , ν 2 , . . . , ν k 2 ) , k 2 k 1 , we introduce a metric by ρ ( x, y ) = ( k 1 i =1 | ξ i ν i | p + k 2 j = k 1 +1 | ν j | p ) 1 /p , 1 p < . Show that p is not a complete metric space. Hint: You have to show that there exists a Cauchy sequence in p , whose limit is not in p ; consider, for instance, the sequence: x 1 = { 1 } , x 2 = { 1 , 1 2 } , x 3 = { 1 , 1 2 , 1 2 2 } , . . . , x n = { 1 , 1 2 , . . . , 1 2 n 1 } , . . . 12. Let f : C [0 , 2] R be a functional defined by f ( x ) = max 0 t 2 x ( t ) (a) Show that f is continuous . (b) Is f uniformly continuous? 13. As discussed in class, the Banach space L p [ a, b ] is separable , for 1 p < and for any finite interval [ a, b ] (see also Example 4, on page 43 of the text). This result does not hold, however, if the interval is infinite, that is ( −∞ , ), and the norm adopted is x = ( lim T →∞ 1 T T T | x ( t ) | p dt ) 1 p

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We denote the space in this case by L p ( −∞ , ). Construct a counterexample to show that L 2 ( −∞ , ) is not separable. Hint : Try the continuum of elements { sin αt, α ( −∞ , ) } as a candidate to generate a counterexample.
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