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frechet_derivative_chain_rule [2013/09/15 20:15] nikolaj |
frechet_derivative_chain_rule [2013/09/15 20:26] nikolaj |
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| ==== Theorem ==== | ==== Theorem ==== | ||
| | @#88DDEE: $X,Y,Z$ ... Banach spaces with topology | | | @#88DDEE: $X,Y,Z$ ... Banach spaces with topology | | ||
| - | | @#88DDEE: $F\in C(X,Y)$ | | + | | @#88DDEE: $f\in C(X,Y)$ | |
| - | | @#88DDEE: $G\in C(Y,Z)$ | | + | | @#88DDEE: $g\in C(Y,Z)$ | |
| - | | @#55EE55: $ D(G\circ F)=(DG)\circ F\ \cdot\ DF $ | | + | | @#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. | ||
