3-hw-116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #3 Last modified: October 9, 2007 Due Thursday, Oct. 11 at class This problem set has been shortened to allow more time for watching baseball playoffs. 1. Showing that a norm topology is consistent with axioms of general topology. (a) Luenberger, section 2.16, problem 6. (b) Invent a collection of open intervals whose intersection is the closed interval [0,1]. (c) Luenberger, section 2.16, problem 7. (d) Invent a collection of closed intervals whose union is the open interval (0,1). 2. Luenberger, section 2.16, problem 8. Remember how to interpret “smallest” in terms of “intersection.” 3. Luenberger, section 2.16, problem 12. 4. Provide the “omitted” proof of Theorem 3 on p. 32. Don’t worry about the Lebesgue integrals: just assume that any reasonable property of the integrals is true. Most of the work has already been done. All you have to do is replace sums by integrals. Remember to take the absolute value of
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3-hw-116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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