ProofListforFinal

# ProofListforFinal - MATHEMATICS 116 FALL 2007-8 CONVEXITY...

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Unformatted text preview: MATHEMATICS 116, FALL 2007-8 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Proof List for the Final Examination Last modified: January 13, 2008 The exam will contain one proof chosen at random from each set of three: 1-3, 4-6, .... 1. Prove that the interior of a convex set C is convex. 2. Given the identity (for a,b > 0) a λ b 1- λ ≤ λa + (1- λ ) b, prove that if p,q > 0 satisfy 1 p + 1 q = 1 and x = { ξ 1 ,ξ 2 , ···} ∈ l p , y = { η 1 ,η 2 , ···} ∈ l q , then ∞ X i =1 | ξ i || η i | ≤ || x || p || y || q . 3. Note: this is finite-dimensional! Prove that the space of sequences x = { ξ 1 ,ξ 2 , ··· ,ξ n } , with the norm || x || = ( n X i =1 | ξ i | p ) 1 p , is a Banach space. 4. Suppose that K is a convex set in a Hilbert space H and x is an element of H that lies outside K . Let k be the vector in K that is closest to x . Prove that k is unique and that ( x- k | k- k ) ≤ 0 for all k ∈ K ....
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ProofListforFinal - MATHEMATICS 116 FALL 2007-8 CONVEXITY...

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