hw1(6) - Homework assignment 1 The first problem assumes...

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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 ² > 0, there is some N such that if n > N , then || f n ( x ) - f ( x ) || < ² , 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, δ > 0 : h B δ ( x ) f n ( x + h ) - f n ( x ) = Z 1 0 Df n ( x + th ) hdt, where B δ ( x ) denotes the open ball of radius δ centered at x . This is an application of Proposition 1 . 167. Show that this implies that the above equality remains valid if we replace f n by f and Df n by g . Conclude that Df = g . Note : Proposition 1 . 166 may be useful.
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