6-HahnBanach

6-HahnBanach - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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Unformatted text preview: MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Outline #6 (Applications of the Hahn-Banach Theorem) Last modified: November 20, 2007 Reading. Luenberger, Chapter 5, sections 5.5 - 5.9 and 5.11-5.13. None of this is relevant to the quiz on Thursday, November 8. Lecture topics. 1. The norm as an all-purpose 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 real-valued (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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6-HahnBanach - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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