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ProofList2

ProofList2 - K then p x 1 x 2 ≤ p x 1 p x 2 7 Let M be a...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Proof List for Quiz 2 on November 8 Last modiﬁed: November 4, 2007 1. Prove that the Banach space l p for 1 p < is separable. 2. Suppose that X is a Hilbert space and that M is a closed subspace of X . Let x be a vector in X and let m 0 be the closest vector to x in M . Prove that m 0 exists and is unique. 3. Prove that the basis { 1 ,t,t 2 , ···} for L 2 [ - 1 , 1] is complete by showing that no function f ( t ) that is orthogonal to all polynomials can exist. You may take it as proved that any such f ( t ) would have a continuous antiderivative F ( t ). 4. Suppose that K is a convex set in a Hilbert space H and x is an element of H 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. Prove that a linear functional f on a normed space X is continuous if and only if it is bounded. 6. Prove that if p ( x ) is the Minkowski functional of a convex set
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Unformatted text preview: K , then p ( x 1 + x 2 ) ≤ p ( x 1 ) + p ( x 2 ). 7. Let M be a closed subspace of a normed vector space X , let p ( x ) be a sublinear functional deﬁned on all of 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 from the extension form of the theorem. You may take it as proved that the Minkowski functional of a convex set is sublinear. 1...
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