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Unformatted text preview: ECE 580 / Math 587 SPRING 2011 Correspondence # 17 April 21, 2011 Solution Set 6 41. (i) We are given that ( x n − x o ,x ) → 0 (and equivalently that ( x,x n − x o ) → 0) for all x ∈ X , and that ∥ x n ∥ → ∥ x ◦ ∥ . Then, ∥ x n − x o ∥ 2 = ∥ x n ∥ 2 + − ( x n ,x o ) − ( x o ,x n − x o ) = ∥ x n ∥ 2 − ∥ x o ∥ 2 − ( x n − x o ,x o ) − ( x o ,x n − x o ) → (ii) Take x o = 0, without any loss of generality. By weak convergence, for every y ∈ X , ( x n ,y ) → 0 as n → ∞ , which means that given y ∈ X , we can find N > 0 such that  ( x n ,y )  < δ for all n > N . Clearly, also, given y 1 ,y 2 ,...,y m ∈ X and δ > 0 there exists N > 0 such that  ( x n ,y i )  < δ for all n > N , i = 1 ,...,m . Now, given the sequence { x n } , choose a subsequence { x n k } as follows: Choose x n 1 = x 1 . Choose x n 2 such that  ( x n 1 ,x n 2 )  < 1 (such an x n 2 exists from the weak convergence property above). Choose x n 3 such that  ( x n 1 ,x n 3 )  < 1 2 ,  ( x n 2 ,x n 3 )  < 1 2 (again use the weak convergence property above, with y 1 = x n 1 , y 2 = x n 2 , and δ = 1 2 ). Iteratively pick x n 1 ,x n 2 ,...,x n k , and choose x n k +1 such that  ( x n i ,x n k +1 )  < 1 k , i = 1 , 2 ,...,k Also, since ( x n ,x ) → 0 for all x ∈ X , ( x n ,x n ) = ∥ x n ∥ 2 can be uniformly bounded, say by M 2 . Then, ∥ y m ∥ 2 = ∥ 1 m m ∑ k =1 x n k ∥ 2 ≤ ( 1 m ) 2 mM 2 + 2 m ∑ i =2 i − 1 ∑ j =1 ( x n j ,x n i ) ≤ ( 1 m ) 2 ( mM 2 + 2( m − 1) ) → 0 as m → ∞ Hence, y m converges strongly to 0. 42. (i) We want to show that a linear functional f on a normed space X can be expressed in the form f ( x ) = < x,x ∗ > , with x ∗ ∈ X ∗ , if and only if it is weakly continuous. First let f ( x ) = < x,x ∗ > , and in the definition of weak continuity given ϵ > 0 choose δ = ϵ and x ∗ 1 = x ∗ . Then,  < x,x ∗ 1 >  < δ ⇒  f ( x )  < ϵ and hence f is weakly continuous at x = θ and thereby everywhere (since f is linear). Note that weak continuity is a stronger notion of continuity than regular continuity, in the sense that weak continuity implies continuity in norm, but not vice versa....
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 Spring '08
 Staff
 Hilbert space, η, Dual space, Topological vector space, Linear functional, xn − xo

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