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Unformatted text preview: MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Proof List for Quiz 3 on December 6 Last modified: December 3, 2007 As usual, the quiz will have one proof chosen at random from the first four and another chosen at random from the last four. I have made simplifying assumptions for some of the longer proofs. If it is not clear what you are allowed to assume, send email to get a clarification. 1. You may take the extension version of the Hahn-Banach theorem as proved. Show that any norm || x || is a sublinear function, then, given a bounded linear functional f defined on subspace M X , with norm || f || M , prove that it can be extended to a bounded linear functional F , with the same norm, defined on all of X . 2. (This is just an important step in the proof of the Riesz representation the- orem. You may use the Hahn-Banach theorem and the Hoelder inequality) Start with f , a bounded linear functional on C [ a,b ] , and let F denote its extension to a functional on...
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- Fall '09