Unformatted text preview: X can be written as the intersection of countably many open subsets of X . 3. [10 points] Let ( X,d ) be a complete metric space and f : X → X a continuous function such that for some integer N > 0, f N = f ◦ f ◦ ··· ◦ f  {z } N times , is a contraction on X . Prove that there exists unique x ∈ X such that f ( x ) = x . 4. [10 points] Consider Banach space C ([0 , 1]). Let F : C ([0 , 1]) → C ([0 , 1]) be a function deﬁned by F ( f ) = sin( f ) . In other words, ( F ( f ))( t ) = sin( f ( t )) for t ∈ [0 , 1]. Prove that F is a continuous function. 1...
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This document was uploaded on 02/22/2011.
 Spring '09
 Math

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