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Unformatted text preview: Math 301 Final Exam, 2005 Prof. Peter Jones December 13, 2005 Try to give as much detail as possible in your answers. If you use a theorem, state clearly which theorem you are using and show that the hypotheses of that theorem apply. 1. a) State the HeineBorel Theorem for ℝ 2 . (Don’t forget to state for which sets it holds!) b) Prove the HB Theorem for the set [0,1] 2 . (This is the closed “unit square.”) 2. Let M be a metric space with metric ρ . In this problem you are not allowed to assume that the metric space M is ℝ . (x – y  = points off!) a) Define what it means for a mapping F to be a contraction on M. (Make sure to have all quantifiers in the correct order!) b) State the Contraction Mapping Principle (CMP) for metric spaces. ( Don't forget any of the hypotheses. Also refer to part a! ) c) Suppose λ ≠ 1 is a complex number, and suppose m,n ∈ ℕ with m < n....
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 Fall '07
 TrietLe
 Math, Topology, #, Continuous function, Metric space

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