245Cfin - Final Exam, Math 245C, June 4, 2007. Exams are...

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Final Exam, Math 245C, June 4, 2007. Exams are due June 13 at 6:00 PM in my mailbox. You may use your course notes or any book available, including Folland and H¨ ormander, but you should write out a complete solution of each problem. J. Garnett 1. A topological space X with topology T is metrizable if there is a metric d on X such that U ∈ T if and only if for all x U there is r > 0 such that B ( x,r ) = { y X : d ( x,y ) < r } ⊂ U. Let X be a compact Hausdor± space and let C ( X ) be the Banach space of continuous real-valued functions on X with the norm || f || = sup {| f ( x ) | : x X } . Prove that C ( X ) is separable (i.e. has a countable dense subset) if and only if X is metrizable. 2. Let X be a real Banach space, let M be a closed subspace of X and let L : M R be a continuous linear functional on M with || L || M * = sup {| L ( x ) | : x M, || x || = 1 } = 1 . Prove that there is a
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245Cfin - Final Exam, Math 245C, June 4, 2007. Exams are...

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