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hw3 - Y is ±nite-dimensional one can choose x so that d...

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ECE 580, Optimization by Vector Space Methods Assignment # 3 Issued: February 6 Due: February 20, 2008 Reading Assignment: Luenberger , complete Chapter 3, and begin Chapter 5. Problems: 10 Before completing this exercise take a look at the Riesz Lemma See for example http://en.wikipedia.org/wiki/Riesz’s_lemma (i) Let Y be a proper closed subset of the normed linear space X (not necessarily a inner-product space). Then, given ε > 0 there exists x X satisfying bardbl x bardbl = 1 and d ( x,Y ) 1 ε . Furthermore, if Y is finite-dimensional one can choose
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Unformatted text preview: Y is ±nite-dimensional one can choose x so that d ( x,Y ) = 1. What if X is a Hilbert space ? (ii) Let X be an in±nite dimensional normed linear space. Show that there is a sequence { x n } ⊂ X satisfying b x b = 1 and b x n − x m b ≥ 1 for each n,m . (iii) Conclude that the closed unit ball is not compact if X is in±nite-dimensional. 11 Luenberger Prob. 3.7 12 Luenberger Prob. 3.12 13 Luenberger Prob. 3.21 14 Luenberger Prob. 3.22...
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