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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 Fall '09
 Math, Topology, Luenberger, Minkowski inequality

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