Lecture08 - Lecture 8 Example: Recall that if F : U Y and G...

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Unformatted text preview: Lecture 8 Example: Recall that if F : U Y and G : U Z are differentiable at a U , then so is H = ( F,G ) : U Y Z and, further, H a ( h ) = ( F a ( h ) ,G a ( h )) . Conversely, if H is differentiable at a then so are F and G . [Let P : Y Z Y be first-factor projection: P ( y,z ) = y . This is lin- ear and bounded (if Y Z has the norm given by, say, k ( y,z ) k = k y k k z k then k P ( y,z ) k k ( y,z ) k ); Thus, P L ( Y Z,Y ) and so P is differentiable, with P ( b,c ) = P for b Y,c Z . Now F = P H is differentiable at a by the chain rule and F a ( h ) = P ( F ( a ) ,G ( a )) H a ( h ) = P ( H a ( h )) A similar argument applies to G .] Consequence: a map F : R m R n is differentiable at a R m iff its components are all differentiable at a . F ( a ) = ( f 1 ( a ) ,...,f n ( a )) . Interlude: Operator norm on M n R . Let X be a Hilbert space. The adjoint of A L ( X ) is A * L ( X ) given by x,y X ( y | A * x...
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Lecture08 - Lecture 8 Example: Recall that if F : U Y and G...

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