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Unformatted text preview: Homework assignment 1 * September 16, 2005 The first problem assumes some knowledge of integration in Banach spaces, a topic we skipped in class (p. 100 105 in the text). 1. Let X and Y be Banach spaces and U open in X . Let { f n } be a sequence of continuously differentiable functions f n : U Y and { Df n } the corresponding sequence of derivatives. Assume that both { f n } and { Df n } converge uniformly to f , and g respectively. ( { f n } converges uniformly to f if for all > 0, there is some N such that if n > N , then  f n ( x ) f ( x )  < , for all x U ). Prove that (a) The limits f and g are continuous. (b) f is differentiable and Df = g . Hint : Use the fact that for all n , there holds that x U, > 0 : h B ( x ) f n ( x + h ) f n ( x ) = Z 1 Df n ( x + th ) hdt, where B ( x ) denotes the open ball of radius centered at x . This is an application of Proposition 1 . 167. Show that this implies that the above equality remains valid if we167....
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 Summer '06
 DeLeenheer

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