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4-hw116

4-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #4 Last modified: October 12, 2007 Due Thursday, Oct. 18 at class This is short enough that you might have time left over to watch the baseball playoffs. 1. (a) Show that the function f ( x ) = 1 x is continuous on (0,1], but not uni- formly continuous. (b) Now consider any continuous functional f : X R whose domain includes a compact subset S of a normed vector space X . Show that if f is continuous on S , it is uniformly continuous. Use no property of compactness but the definition (Luenberger, p. 40). The example in part (a) used an interval that was not compact. This is a famous (named) theorem, so if you get stuck there is plenty of help available online! 2. (a) After consulting at least two sources, give a careful statement of the “well-known Weierstrass approximation theorem” to which Luenberger alludes on page 42. Does the function being approximated have to be uniformly continuous, or just continuous, or does it not matter?

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4-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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