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smooth_function [2013/09/13 20:34] nikolaj |
smooth_function [2014/12/04 16:39] nikolaj |
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| ===== Smooth function ===== | ===== Smooth function ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $X,Y$ ... Banach spaces with topology | | + | | @#55CCEE: context | @#55CCEE: $X,Y$ ... Banach spaces with topology | |
| - | + | | @#FFBB00: definiendum | @#FFBB00: $f\in C^\infty(X,Y)$ | | |
| - | | @#FFBB00: $f\in C^\infty(X,Y)$ | | + | | @#FFFDDD: for all | @#FFFDDD: $k\in \mathbb N$ | |
| - | + | | @#55EE55: postulate | @#55EE55: $f\in C^k(X,Y)$ | | |
| - | | @#55EE55: $\forall (k\in \mathbb N).\ f\in C^k(X,Y)$ | | + | | @#55EE55: postulate | @#55EE55: $\lim_{m\to\infty}D^mf$ ... continuous | |
| - | | @#55EE55: $\lim_{k\to\infty}D^kf$ ... continuous | | + | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| + | === Formalities === | ||
| + | Here $D^mf$ denotes the $m$th component of the sequence $Df, D^2f, D^3f,\dots$. | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Differentiable_function|Differentiable function]], [[http://en.wikipedia.org/wiki/Smooth_function|Smooth function]] | Wikipedia: [[http://en.wikipedia.org/wiki/Differentiable_function|Differentiable function]], [[http://en.wikipedia.org/wiki/Smooth_function|Smooth function]] | ||
| ==== Parents ==== | ==== Parents ==== | ||
| - | === Requirements === | + | === Context === |
| [[Differentiable function]], [[Limit in a metric space]] | [[Differentiable function]], [[Limit in a metric space]] | ||
