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...
View
Full Document
 Spring '09
 Math, Topology, Continuous function, Empty set, Metric space, Open set, Topological space

Click to edit the document details