ProofList1

ProofList1 - l q , then X i =1 | i || i | || x || p || y ||...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Proofs to learn for Quiz 1 on October 11. Two of these, chosen at random, will appear on the quiz. Last modified: October 9, 2007 1. Prove that the interior of a convex set C is convex. 2. Prove that the closure of a convex set C is convex. 3. Prove that if a vector space V is generated by the set of k vectors { v 1 ,v 2 , ··· ,v k } , any set of k +1 vectors { w 1 ,w 2 , ··· ,w k +1 } in V must be linearly dependent. 4. For a vector space with arbitrary norm || x || , prove that if the sequence { x n } converges to x and the sequence { x n } also converges to y , then x = y . 5. 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 2 , ···} ∈ l p , y = { η 1 2 , ···} ∈
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Unformatted text preview: l q , then X i =1 | i || i | || x || p || y || q . 1 6. Note: this is nite-dimensional! Given the Holder inequality: if p,q > 0 satisfy 1 p + 1 q = 1 and x = { 1 , 2 , , n } l p , y = { 1 , 2 , , n } l q , then n X i =1 | i || i | || x || p || y || q , prove the Minkowski inequality || x + y || p || x || p + || y || p . 7. Prove that every convergent sequence is a Cauchy sequence. 8. Note: this is nite-dimensional! Prove that the space of sequences x = { 1 , , n } , with the norm || x || = ( n X i =1 | i | p ) 1 p , is a Banach space. 2...
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ProofList1 - l q , then X i =1 | i || i | || x || p || y ||...

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