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8-hw116

# 8-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #8 Last modified: November 14, 2007 Due Tuesday, Nov. 20 at class. 1. Luenberger, Problem 5.14.1. 2. Luenberger, Problem 5.14.3. Explain why this result shows that the dual of l cannot be l 1 . Hint: Choose a countable dense subset of the unit sphere in X * : the ele- ments of X * for which || x * || = 1. For each x * n in this subset, choose a vector x n for which | < x n | x * n > | is close to 1 (say bigger than 1 2 ). Now form the closure of the space Y generated by all these x n . Prove (easily) that this space is separable. If it also equals X , you are done. So assume that X contains an element x 0 that is not in Y , and show that this assumption leads to a contradiction. 3. Luenberger, Problem 5.14.2. This is challenging. Both dual spaces are isomorphic to l 1 , but if the isomorphism is different they are “not identical.” For c 0 you can use an approach that parallels the one for l p and l 1 . It is easy to show, using the Hoelder inequality, that any element of l 1 defines

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8-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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