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Unformatted text preview: Analysis Final Sketch Solutions Write solutions in a neat and logical fashion, giving complete reasons for all steps. 1. State what it means for a Banach space X to be (i) uniformly convex (ii) strictly convex . Prove carefully that if X uniformly convex then X is strictly convex. With brief justification, give an example to show that strict convexity does not imply uniform convexity. Solution (i) If ( a n ) and ( b n ) are sequences of unit vectors such that k a n + b n k → 2 then k a n b n k → 0. (ii) Vectors a and b satisfying k a + b k = k a k + k b k are parallel. Uniform ⇒ strict : Let (nonzero) a and b satisfy k a + b k = k a k + k b k . Claim : The unit vectors ˆ a = a/ k a k and ˆ b = b/ k b k satisfy k ˆ a + ˆ b k = 2. Granted this, passage to constant sequences shows (by uniform convexity) that ˆ a and ˆ b are equal, so a and b are parallel. Proof of claim : note first that k ˆ a + ˆ b k = k ( k b k a + k a k b ) k / k a kk b k . Here, k a + b k = k a k + k b k...
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 Fall '09
 Robinson
 Linear Algebra, Vector Space, Hilbert space, Topological vector space, uniformly convex

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