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 frechet_derivative_chain_rule [2013/09/15 20:15]nikolaj frechet_derivative_chain_rule [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/09/15 20:26 nikolaj 2013/09/15 20:15 nikolaj 2013/09/15 20:13 nikolaj 2013/09/15 20:11 nikolaj 2013/09/15 20:11 nikolaj old revision restored (2013/09/15 19:25) Next revision Previous revision 2013/09/15 20:26 nikolaj 2013/09/15 20:15 nikolaj 2013/09/15 20:13 nikolaj 2013/09/15 20:11 nikolaj 2013/09/15 20:11 nikolaj old revision restored (2013/09/15 19:25) Line 1: Line 1: ===== Fréchet derivative chain rule ===== ===== Fréchet derivative chain rule ===== ==== Theorem ==== ==== Theorem ==== - | @#88DDEE: $X,Y,Z$ ... Banach spaces with topology | + | @#55CCEE: context ​    | @#55CCEE: $X,Y,Z$ ... Banach spaces with topology | - | @#88DDEE: $F\in C(X,Y)$ | + | @#55CCEE: context ​    | @#55CCEE: $f\in C(X,Y)$ | - | @#88DDEE: $G\in C(Y,Z)$ | + | @#55CCEE: context ​    | @#55CCEE: $g\in C(Y,Z)$ | - | @#55EE55: $D(G\circ F)=(DG)\circ ​F\ \cdot\ ​DF$ | + | @#55EE55: postulate ​  | @#55EE55: $D(g\circ f)=(Dg)\circ ​f\ \cdot\ ​Df$ | where $\circ$ denotes the concatenation of functions of $X,Y$, which is taken to bind stronger than the concatenation $\cdot$ of linear operators. where $\circ$ denotes the concatenation of functions of $X,Y$, which is taken to bind stronger than the concatenation $\cdot$ of linear operators. Line 17: Line 17: Wikipedia: [[http://​en.wikipedia.org/​wiki/​Chain_rule|Chain rule]], [[http://​en.wikipedia.org/​wiki/​Chain_rule_%28disambiguation%29|Chain rule (disambiguation)]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Chain_rule|Chain rule]], [[http://​en.wikipedia.org/​wiki/​Chain_rule_%28disambiguation%29|Chain rule (disambiguation)]] ==== Parents ==== ==== Parents ==== - === Requirements ​=== + === Context ​=== [[Fréchet derivative]] [[Fréchet derivative]] 