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Lecture12 - Lecture 12 Denition 1 Let U be open in X When F...

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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 0 : U L ( X, Y ) : a 7→ F 0 a and we call F continuously differentiable when F 0 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 0 a is defined for all a U and that F 0 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 0 a ( h 1 , h 2 ) = 1 F a ( h 1 ) + 2 F a ( h 2 ) . Specifically, 1 F a ( h 1 ) = F 0 a ( h 1 , 0) 2 F a ( h 2 ) = F 0 a (0 , h 2 ) . Now consider the maps defined below: α 1 : X 1 X 1 × X 2 : h 1 7→ ( h 1 , 0) α 2 : X 2 X 1 × X 2 : h 2 7→ (0 , h 2 ) . 1
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α 1 and α 2 are linear and k α 1 ( h 1 ) k = k ( h 1 , 0) k = k h 1 k k α 2 ( h 2 ) k = k (0 , h 2 ) k = k h 2 k . It follows that k α 1 k = 1 = k α 2 k . We can see that 1 F : U L ( X 1 , Y ) : a 7→ F 0 a α 1 2 F : U L ( X 2 , Y ) : a 7→ F 0 a α 2 . Since F 0 is continuous at each a let ε > 0 and choose δ such that if b U with || a - b || < δ then k F 0 a - F 0 b k < ε, thus for i = 1 or 2 k i F a - i F b k = k F 0 a α i - F 0 b α i k = k ( F 0 a - F 0 b ) α i k ≤ k F 0 a - F 0 b k k α i k = k F 0 a - F 0 b k < ε. So 1 F and 2 F are continuous at all a U . ( ) Suppose 1 F and 2 F are defined and continuous for all a U . From the last Theorem of “Lecture 11” we have that F is differentiable at each a U .
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