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bounded_function [2013/09/06 22:58] nikolaj created |
bounded_function [2014/03/21 11:11] (current) |
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| ===== Bounded function ===== | ===== Bounded function ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $ \langle X,d\rangle $ ... metric space | | + | | @#55CCEE: context | @#55CCEE: $ \langle X,d\rangle $ ... metric space | |
| - | | @#FFBB00: $f\in \text{BoundedFunction}(X)$ | | + | | @#FFBB00: definiendum | @#FFBB00: $f\in \text{BoundedFunction}(X)$ | |
| - | | @#55EE55: $d:X\times X \to \mathbb{R}_+$ | | + | | @#DDDDDD: range | @#DDDDDD: $x,a\in X$ | |
| + | | @#DDDDDD: range | @#DDDDDD: $M\in \mathbb R$ | | ||
| - | | @#DDDDDD: $x,a\in X$ | | + | | @#55EE55: postulate | @#55EE55: $ \exists a, M.\ \forall x.\ d(f(x),a) < M$ | |
| - | | @#DDDDDD: $M\in \mathbb R$ | | + | |
| - | + | ||
| - | | @#55EE55: $ \exists a, M.\ \forall x.\ d(f(x),a) < M$ | | + | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| + | For linear operators between normed vector spaces, being bounded for each input element and continuous coincide. | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Bounded_function|Bounded function]] | Wikipedia: [[http://en.wikipedia.org/wiki/Bounded_function|Bounded function]] | ||
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| === Subset of === | === Subset of === | ||
| [[Function]] | [[Function]] | ||
| - | === Requirements === | + | === Context === |
| [[Metric]] | [[Metric]] | ||
