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linear_approximation [2014/02/20 18:43] nikolaj old revision restored (2014/02/13 16:13) |
linear_approximation [2014/04/11 16:42] nikolaj |
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| ===== Linear approximation ===== | ===== Linear approximation ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#88DDEE: $X,Y$ ... Banach spaces with topology | | + | | @#55CCEE: context | @#55CCEE: $X,Y$ ... Banach spaces with topology | |
| - | | @#88DDEE: $\mathcal O$ ... open in $X$ | | + | | @#55CCEE: context | @#55CCEE: $\mathcal O$ ... open in $X$ | |
| - | | @#88DDEE: $x\in\mathcal O$ | | + | | @#55CCEE: context | @#55CCEE: $x\in\mathcal O$ | |
| - | | @#88DDEE: $f:\mathcal O\to Y$ | | + | | @#55CCEE: context | @#55CCEE: $f:\mathcal O\to Y$ | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $J_x^f$ | | |
| - | | @#FFBB00: $J_x^f$ | | + | | @#55EE55: postulate | @#55EE55: $J_x^f$ ... bounded linear operator from $X$ to $Y$ | |
| - | + | | @#55EE55: postulate | @#55EE55: $\mathrm{lim}_{h\to 0}\ \Vert f(x+h)-f(x)-J_x^f(h)\Vert / \Vert h\Vert\ =\ 0$ | | |
| - | | @#55EE55: $J_x^f$ ... bounded linear operator from $X$ to $Y$ | | + | |
| - | | @#55EE55: $\mathrm{lim}_{h\to 0}\ \Vert f(x+h)-f(x)-J_x^f(h)\Vert / \Vert h\Vert\ =\ 0$ | | + | |
| ==== Discussion ==== | ==== Discussion ==== | ||
