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# weak - FUNCTIONAL ANALYSIS LECTURE NOTES WEAK AND WEAK...

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FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE CHRISTOPHER HEIL 1. Weak and Weak* Convergence of Vectors Definition 1.1. Let X be a normed linear space, and let x n , x X . a. We say that x n converges , converges strongly , or converges in norm to x , and write x n x , if lim n →∞ bardbl x x n bardbl = 0 . b. We say that x n converges weakly to x , and write x n w x , if μ X , lim n →∞ ( x n ) = ( x,μ ) . Exercise 1.2. a. Show that strong convergence implies weak convergence. b. Show that weak convergence does not imply strong convergence in general (look for a Hilbert space counterexample). If our space is itself the dual space of another space, then there is an additional mode of convergence that we can consider, as follows. Definition 1.3. Let X be a normed linear space, and suppose that μ n , μ X . Then we say that μ n converges weak* to μ , and write μ n w* −→ μ , if x X, lim n →∞ ( x,μ n ) = ( x,μ ) . Note that weak* convergence is just “pointwise convergence” of the operators μ n ! Remark 1.4. Weak* convergence only makes sense for a sequence that lies in a dual space X . However, if we do have a sequence { μ n } n N in X , then we can consider three types of convergence of μ n to μ : strong, weak, and weak*. By definition, these are: μ n μ ⇐⇒ lim n →∞ bardbl μ μ n bardbl = 0 , μ n w μ ⇐⇒ T X ∗∗ , lim n →∞ ( μ n ,T ) = ( μ,T ) , μ n w* −→ μ ⇐⇒ x X, lim n →∞ ( x,μ n ) = ( x,μ ) . c circlecopyrt 2007 by Christopher Heil. 1

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2 WEAK AND WEAK* CONVERGENCE Exercise 1.5. Given μ n , μ X , show that μ n μ = μ n w μ = μ n w* −→ μ. (1.1) If X is reflexive, show that μ n w μ ⇐⇒ μ n w* −→ μ. In general, however, the implications in (1.1) do not hold in the reverse direction. Lemma 1.6. a. Weak* limits are unique. b. Weak limits are unique. Proof. Suppose that X is a normed linear space, and that we had both μ n w* −→ μ and μ n w* −→ ν in X . Then, by definition, x X, ( x,μ ) = lim n →∞ ( x,μ n ) = ( x,ν ) , so μ = ν . b. Suppose that we have both x n w x and x n w y in X . Then, by definition, μ X , ( x,μ ) = lim n →∞ ( x n ) = ( y,μ ) .
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weak - FUNCTIONAL ANALYSIS LECTURE NOTES WEAK AND WEAK...

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