Lecture08

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

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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

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

View Full Document
Ask a homework question - tutors are online