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Unformatted text preview: constant K > 0 such that  f ( x )f ( y )  K  xy  for all x,y [0 , 1]. Denote by Lip([0 , 1]) the collection of all Lipschitz functions on [0 , 1]. It is trivial to show that Lip([0 , 1]) is a vector space with respect to the usual operations 1 of addition and scalar multiplications of functions (you do not need to write the proof). For f Lip([0 , 1]) set k f k =  f (0)  + sup x,y 1 x 6 = y  f ( x )f ( y )   xy  . (1) Show that k k is a norm on C ([0 , 1]). (2) Show that (Lip([0 , 1]) , k k ) is a Banach space. 5. [10 points] Let ( X,d ) be a complete metric space and E n , n = 1 , 2 , , a sequence of closed sets such that diam( E n ) = sup { d ( x,y ) : x,y E n } < . If E n E n +1 for all n , and if lim n diam( E n ) = 0, prove that n =1 E n consists of exactly one point. 2...
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This document was uploaded on 02/22/2011.
 Spring '09
 Integers

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