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5-DualSpace

# 5-DualSpace - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Outline #5 (Dual Spaces and the Hahn-Banach Theorem) Last modified: November 1, 2007 Reading. Luenberger, Chapter 5, sections 5.1 - 5.4 and section 5.12. These sections of Chapter 5 are fair game for the quiz on Thursday, November 8. Lecture topics. 1. Linear functionals A transformation (function) f is called a functional if its codomain is the real numbers. It is a linear functional if f ( αx + βy ) = αf ( x ) + βf ( y ) In a Hilbert space H , any vector y determines a linear function by the rule f y ( x ) = ( y | x ). What functional on E 3 is determined by y = (1 , 1 , 1)? What functional on l 2 is determined by y = (1 , 1 2 , 1 3 , · · · )? What functional on L 2 [0 , 1] is determined by y = e t - 1 ? Remember that p = 2 was a special case for the Hoelder inequality, because q = 2 also. Why, in this case, does the Hoelder inequality reduce to a statement about inner products? 1

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If x = ξ 1 , ξ 2 , · · · ∈ l 4 3 , does X i =1 1 i ξ i converge? Can it be identified with an element of l 4 3 ? 2. Continuity of linear functionals To understand why you have probably never thought about this issue, prove that any linear functional on a finite-dimensional normed vector space is continuous. This is not entirely trivial, since the norm is arbitrary. To understand why there is a problem, show that on l 2 , the linear functional f ( x ) = X k =1 k is discontinuous at x = θ . 2
Prove that if a linear functional f is continuous at one point x 0 on a normed space X , it is continuous at any other point x . The key theorem: on a normed space X , a linear functional X is continuous if and only if it is bounded. Prove that bounded implies continuous. Prove that continuous implies bounded. 3. Making linear functionals into vector space Forgetting “normed” for the moment, how do we define operations that turn linear functionals into elements of a vector space (the “algebraic dual”)? 3

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4. Introducing a norm into the dual space Given any x , we can form two real numbers, the norm || x || and the absolute value | f ( x ) | . For continuous (bounded) linear functionals, the norm of f is defined to be the “greatest possible ratio.” This leads to several equivalent formulas for || f || , all useful from time to time. In terms of the ratio of | f ( x ) | to || x || In terms of vectors of norm 1 (on the unit sphere) In terms of vectors of norm 1 (in the unit ball) As the greatest lower bound of a set Prove that all the requirements of a norm are met. 4
5. The normed dual Let X be a normed linear vector space (perhaps not a Banach space). A functional f or g need not be linear. Even if linear, it may be unbounded. The bounded linear functionals on X form a normed vector space X * , the dual space. To indicate clearly that a functional is an element of X * , we give it a name like x * and denote the value of x * at x by x * ( x ) or < x, x * > .

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5-DualSpace - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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