Unformatted text preview: f ( x ) ± c g . 6. Lipschitz Equivalent Theorem 10.8 on page 107 of de la Fuente says that all norms on R n are Lipschitz-equivalent to the Euclidean norm. The Theorem is correct, but is the proof correct? (a) Suppose q µ q : ( R n ;d ) ! ( R + ;& ) is a norm on R n . d is the metric generated by the norm, d ( x;y ) = q x & y q . & is the Euclidean metric. Show that q µ q is a continuous function. (Hint: Use the triangle inequality.) (b) Now consider the Euclidean norm q µ q E : R n ! R + . The unit circle on R n is de&ned as C = f x 2 R n : q x q E = 1 g . Show that C is compact. (Hint: Show that C is closed and bounded.) (c) Can we use the result of part a and the extreme-value theorem to prove that that q µ q attains a minimum and a maximum in the set C de&ned in part b? 1...
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This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at Berkeley.
- Summer '08