ProofList2-Clarifications

# ProofList2-Clarifications - X and let f m be a linear...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Improved statements of proofs for Quiz 2 Last modiﬁed: November 7, 2007 Notes in parentheses were not added to the text used for the quiz. They address issues that were raised in emails. 3 (This is in Luenberger and was done in class.) 4 Suppose that K is a closed convex set in a Hilbert space H and that x is an element of X that lies outside K . Let k 0 be the vector in K that is closest to x . Prove that k 0 is unique and that ( x - k 0 | k - k 0 ) 0 for all k K . 5 (If you want to use the fact that a linear functional that is continuous at one point is continuous everywhere, you must prove it.) 7 Let M be a closed subspace of a normed vector space X , let p ( x ) be a continuous sublinear functional deﬁned on all of
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Unformatted text preview: X , and let f ( m ) be a linear functional deﬁned on M that satisﬁes f ( m ) ≤ p ( m ) for all m ∈ M . Let y be a vector in X that is not in M . Prove that it is possible to deﬁne a linear functional g on the subspace [ M + y ] such that g ( x ) ≤ p ( x ) for all x ∈ [ M + y ]. 8 State the geometric version of the Hahn-Banach theorem, draw a diagram to illustrate it for the special case of a zero-dimensional linear variety in R 2 , and prove it by using the extension form of the theorem. You may take it as proved that the Minkowski functional of a convex set is sublinear, continuous, non-negative, and ﬁnite. 1...
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