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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 firstfactor 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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 Summer '10
 Staff
 Linear Algebra, Algebra, Hilbert space, Compact space, A∗, uA∗ Au

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