Lecture12 - Lecture 12 Definition 1. Let U be open in X ....

Info iconThis preview shows pages 1–3. 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: Lecture 12 Definition 1. Let U be open in X . When F : U Y is differentiable at all points a U , its derivative is the map F : U L ( X,Y ) : a 7 F a and we call F continuously differentiable when F is continuous. Theorem 1. Suppose U contained in X 1 X 2 ; then F : U Y is continu- ously differentiable if and only if 1 F and 2 F are defined and continuous at each point of U . Proof. Scribal Exercise For any a = ( a 1 ,a 2 ) U , let || a || = max {|| a 1 || , || a 2 ||} . ( ) First, suppose that F is continuously differentiable. This means that F a is defined for all a U and that F is continuous at a . By the last theorem of Lecture 9 1 F a and 2 F a exist at every a U and F a ( h 1 ,h 2 ) = 1 F a ( h 1 ) + 2 F a ( h 2 ) . Specifically, 1 F a ( h 1 ) = F a ( h 1 , 0) 2 F a ( h 2 ) = F a (0 ,h 2 ) . Now consider the maps defined below: 1 : X 1 X 1 X 2 : h 1 7 ( h 1 , 0) 2 : X 2 X 1 X 2 : h 2 7 (0 ,h 2 ) . 1 1 and 2 are linear and k 1 ( h 1 ) k = k ( h 1 , 0) k = k h 1 k k 2 ( h 2 ) k = k (0 ,h 2 ) k = k h 2 k . It follows that k 1 k = 1 = k 2 k . We can see that 1 F : U L ( X 1 ,Y ) : a 7 F a 1 2 F : U L ( X 2 ,Y ) : a 7 F a 2 . Since F is continuous at each a let > 0 and choose such that if b U with || a- b || < then k F a- F b k < , thus for i = 1 or 2 k i F a- i F b k = k F a i- F b i k = k ( F a- F b ) i k k F a- F b kk i k = k F a- F b k < ....
View Full Document

This note was uploaded on 07/08/2011 for the course MAT 6932 taught by Professor Staff during the Summer '10 term at University of Florida.

Page1 / 8

Lecture12 - Lecture 12 Definition 1. Let U be open in X ....

This preview shows document pages 1 - 3. Sign up to view the full document.

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