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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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 Spring '09
 Integers, 1 K, Metric space, Cauchy, complete metric space, usual sup norm, 0 1 k

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