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 bounded_function [2013/09/06 23:00]nikolaj bounded_function [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/09/06 23:03 nikolaj 2013/09/06 23:00 nikolaj 2013/09/06 22:58 nikolaj created Next revision Previous revision 2013/09/06 23:03 nikolaj 2013/09/06 23:00 nikolaj 2013/09/06 22:58 nikolaj created Line 1: Line 1: ===== 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: $f\in \text{BoundedFunction}(X)$ | - | @#DDDDDD: $x,a\in X$ | + | @#DDDDDD: range       | @#DDDDDD: $x,a\in X$ | - | @#DDDDDD: $M\in \mathbb R$ | + | @#DDDDDD: range       | @#DDDDDD: $M\in \mathbb R$ | - | @#55EE55: $\exists a, M.\ \forall x.\ d(f(x),a) < M$ | + | @#55EE55: postulate ​  | @#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]] Line 16: Line 17: === Subset of === === Subset of === [[Function]] [[Function]] - === Requirements ​=== + === Context ​=== [[Metric]] [[Metric]] 