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bounded_linear_operator [2013/09/13 21:51] nikolaj |
bounded_linear_operator [2016/07/31 12:42] nikolaj |
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===== Bounded linear operator ===== | ===== Bounded linear operator ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#88DDEE: $V,W$ ... normed vector spaces | | + | | @#55CCEE: context | @#55CCEE: $V,W$ ... normed vector spaces | |
- | + | | @#FFBB00: definiendum | @#FFBB00: $A\in\mathrm{BoundedLinOp}(V,W)$ | | |
- | | @#FFBB00: $A\in\mathrm{BoundedLinOp}(V,W)$ | | + | | @#55EE55: postulate | @#55EE55: $A\in\mathrm{Hom}(V,W)$ | |
- | + | | @#DDDDDD: range | @#DDDDDD: $M\in\mathbb R, M>0$ | | |
- | | @#55EE55: $A\in\mathrm{Hom}(V,W)$ | | + | |
- | + | ||
- | | @#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: $\exists M.\ \Vert Av\Vert_W\le M\Vert v\Vert_V $ | | + | ----- |
+ | === Discussion === | ||
+ | A linear operator on a metrizable vector space is bounded if and only if it is continuous. | ||
- | ==== Discussion ==== | ||
- | 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]] | ||
- | === Requirements === | + | === Context === |
[[Normed vector space]] | [[Normed vector space]] |