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bounded_linear_operator [2014/03/21 11:11] 127.0.0.1 external edit |
bounded_linear_operator [2016/07/31 12:42] nikolaj |
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| ==== Set ==== | ==== Set ==== | ||
| | @#55CCEE: context | @#55CCEE: $V,W$ ... normed vector spaces | | | @#55CCEE: context | @#55CCEE: $V,W$ ... normed vector spaces | | ||
| - | |||
| | @#FFBB00: definiendum | @#FFBB00: $A\in\mathrm{BoundedLinOp}(V,W)$ | | | @#FFBB00: definiendum | @#FFBB00: $A\in\mathrm{BoundedLinOp}(V,W)$ | | ||
| - | |||
| | @#55EE55: postulate | @#55EE55: $A\in\mathrm{Hom}(V,W)$ | | | @#55EE55: postulate | @#55EE55: $A\in\mathrm{Hom}(V,W)$ | | ||
| - | |||
| | @#DDDDDD: range | @#DDDDDD: $M\in\mathbb R, M>0$ | | | @#DDDDDD: range | @#DDDDDD: $M\in\mathbb R, M>0$ | | ||
| | $v\in V$ | | | $v\in V$ | | ||
| - | |||
| | @#55EE55: postulate | @#55EE55: $\exists M.\ \Vert Av\Vert_W\le M\Vert v\Vert_V $ | | | @#55EE55: postulate | @#55EE55: $\exists M.\ \Vert Av\Vert_W\le M\Vert v\Vert_V $ | | ||
| - | ==== Discussion ==== | + | ----- |
| + | === Discussion === | ||
| A linear operator on a metrizable vector space is bounded if and only if it is continuous. | A linear operator on a metrizable vector space is bounded if and only if it is continuous. | ||
| + | |||
| === Reference === | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Bounded_linear_operator|Bounded linear operator]] | + | Wikipedia: |
| - | ==== Parents ==== | + | [[http://en.wikipedia.org/wiki/Bounded_linear_operator|Bounded linear operator]] |
| + | |||
| + | ----- | ||
| === Subset of === | === Subset of === | ||
| [[Vector space homomorphism]] | [[Vector space homomorphism]] | ||
| === Context === | === Context === | ||
| [[Normed vector space]] | [[Normed vector space]] | ||
