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Unformatted text preview: MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Outline #6 (Applications of the HahnBanach Theorem) Last modified: November 20, 2007 Reading. Luenberger, Chapter 5, sections 5.5  5.9 and 5.115.13. None of this is relevant to the quiz on Thursday, November 8. Lecture topics. 1. The norm as an allpurpose sublinear function  Corollary 1 on page 112. Show that any norm  x  is a sublinear function. 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 . The trick is to choose p ( x ) =  f  M  x  . A really simple example: In E 2 (Euclidean norm), let M be the line x = y and define f ( x ) = 2 x on M . What is the extension F ( x,y ) to all of E 2 ? Show (Corollary 2) for any x ∈ X , how to construct a nonzero bounded linear functional F , defined on all of X , such that  F ( x )  =  F  x  . 1 2. The Dual of C [ a,b ] Recall that C [ a,b ] is the Banach space of realvalued (uniformly) continuous functions on the closed interval [ a,b ] with norm defined by sup  f ( x )  . Here are some examples of linear functionals on C [0 , 1]. • Z 1 x ( t ) dt • Z 1 f ( t ) x ( t ) dt • f ( 1 2 ) • 1 12 ( f (0) + 4 f ( 1 4 ) + 2 f ( 1 2 ) + 4 f ( 3 4 ) + f (1)) • 1 2 f (1) + 1 4 f ( 1 2 ) + 1 8 f ( 1 4 ) + ··· The challenge is to find a way of representing all of these in a unified manner without using Dirac delta functions. This can be done elegantly by using the Stieltjes integral. Let g ( t ) be a bounded function, not necessarily continuous. Let f ( t ) be a continuous function (not a requirement for the Riemann integral, but needed if g ( t ) is discontinuous). Break up the interval so that a = t < t 1 < ··· t n = b . Then the Stieltjes integral is defined by forming, not the usual left Riemann sum n X i =1 f ( t i 1 )( t i t i 1 ) . but the more general n X i =1 f ( t i 1 )( g ( t i ) g ( t i 1 )) . The Stieltjes integral is written as Z b a f ( t ) dg ( t ) . 2 3. Examples of Stieltjes integrals Special cases: What is R 1 f ( t ) dg ( t ) if • g ( t ) = t ? • g ( t ) is differentiable? • g ( t ) = 0 for t < 1 2 ,g ( t ) = 1 for t ≥ 1 2 ? • g ( t ) = t for t < 1 2 ,g ( t ) = t + 2 for t ≥ 1 2 ? • g ( t ) is the probability that random variable T ≤ t ? 3 The Riesz representation theorem (Luenberger, page 113) says that any bounded linear functional f on C [ a,b ] can be expressed as a Stieltjes integral Z b a f ( t ) dv ( t ) However, there is a catch. The function v must be of bounded variation. Loosely speaking, this means that Z b a  dv ( t )  must be finite....
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This document was uploaded on 08/30/2011.
 Fall '09
 Math

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