3-Banach - MATHEMATICS 116, FALL 2007 CONVEXITY AND...

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Unformatted text preview: MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Outline #3 (Banach Spaces) Last modified: October 4, 2007 Reading. Luenberger, Sections 2.10 - 2.15 Lecture topics. 1. Another Cauchy-Schwarz proof The so-called Cauchy-Schwarz inequality was first published by Bunyakovsky. This fact illustrates Stiglers Law of Eponymy: No law, theorem, or discovery is named after its originator. The law applies to itself, since long before Stigler formulated it, A. N. Whitehead noted that, Everything of importance has been said before, by someone who did not discover it. Here is a proof that will generalize to arbitrary norms. By elementary algebra, ( x- y ) 2 xy 1 2 x 2 + 1 2 y 2 . Let a = x 2 ,b = y 2 . a 1 2 b 1 2 1 2 a + 1 2 b. This says that the geometric mean of a and b is less than the arithmetic mean. Now consider vectors x and y , infinite sequences { i } and { i } respectively. Both vectors have finite l 2 norms, where || x || 2 = ( X i =1 2 i ) 1 2 . Use the identity above for each component i , with a = 2 i || x || 2 ,b = 2 i || y || 2 . | i | || x || | i | || y || 1 2 2 i || x || 2 + 1 2 2 i || y || 2 . What do you get by summing over all i ? 1 2. Consider a specific vector x X , the infinite sequence i = 1 i . 3 What is the smallest integer p for which the norm || x || p is finite? If i = 1 i . 3 , does i =1 i i converge? If i = 1 i . 9 , does i =1 i i converge? We can view i as an infinite set of coefficients that define a linear functional on X . A sufficient condition for i =1 i i to converge is that the norm || y || 4 3 should be finite. Work out this norm for the case where i = 1 i . 9 . Cauchy-Schwarz was a special situation, because when we use the l 2 norm in X , we also want the same l 2 norm in the dual space of linear functionals on X . In general, if we use the l p norm in X and want the sum that evaluates a linear functional, i =1 i i , to converge, we should use the l q norm, where 1 p + 1 q = 1, on the dual space of linear functionals. Thus we need to prove the Hoelder inequality, Theorem 1 on page 29. If p,q > 0 satisfy 1 p + 1 q = 1 and x = { 1 , 2 , } l p , y = { 1 , 2 , } l q , then X i =1 | i i | || x || p || y || q . 2 3. Hoelder proof The tricky part is generalizing xy 1 2 ( x + y ). The following approach is equivalent to Luenberger, p. 30, but it suggests a general way of inventing inequalities. Start with the monotone decreasing function t - 1- 1 for 0 < < 1. Sketch its graph. Where does an antiderivative of this function have its maximum?...
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3-Banach - MATHEMATICS 116, FALL 2007 CONVEXITY AND...

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