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bounded_linear_operator [2013/09/13 21:50] nikolaj |
bounded_linear_operator [2016/07/31 12:42] nikolaj |
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| ===== Bounded linear operator ===== | ===== Bounded linear operator ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $X,Y$ ... normed vector spaces | | + | | @#55CCEE: context | @#55CCEE: $V,W$ ... normed vector spaces | |
| + | | @#FFBB00: definiendum | @#FFBB00: $A\in\mathrm{BoundedLinOp}(V,W)$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $A\in\mathrm{Hom}(V,W)$ | | ||
| + | | @#DDDDDD: range | @#DDDDDD: $M\in\mathbb R, M>0$ | | ||
| + | | $v\in V$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $\exists M.\ \Vert Av\Vert_W\le M\Vert v\Vert_V $ | | ||
| - | | @#FFBB00: $A\in\mathrm{BoundedLinOp}(X,Y)$ | | + | ----- |
| + | === Discussion === | ||
| + | A linear operator on a metrizable vector space is bounded if and only if it is continuous. | ||
| - | | @#DDDDDD: $M\in\mathbb R, M>0$ | | + | === Reference === |
| - | | $x\in X$ | | + | Wikipedia: |
| + | [[http://en.wikipedia.org/wiki/Bounded_linear_operator|Bounded linear operator]] | ||
| - | | @#55EE55: $\exists M.\ \Vert Ax\Vert_Y\le M\Vert x\Vert_X $ | | + | ----- |
| - | + | ||
| - | ==== Discussion ==== | + | |
| - | A linear operator on a metrizable vector space is bounded if and only if it is continuous. | + | |
| - | === Reference === | + | |
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Bounded_linear_operator|Bounded linear operator]] | + | |
| - | ==== Parents ==== | + | |
| === Subset of === | === Subset of === | ||
| [[Vector space homomorphism]] | [[Vector space homomorphism]] | ||
| - | === Requirements === | + | === Context === |
| [[Normed vector space]] | [[Normed vector space]] | ||
