hw1(6) - Homework assignment 1 * September 16, 2005 The...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Page1 / 3

hw1(6) - Homework assignment 1 * September 16, 2005 The...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online