Math 301a Final Exam 2005

# Math 301a Final Exam 2005 - Math 301 Final Exam 2005 Prof...

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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 Heine-Borel Theorem for ℝ 2 . (Don’t forget to state for which sets it holds!) b) Prove the H-B 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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Math 301a Final Exam 2005 - Math 301 Final Exam 2005 Prof...

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