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bounded_linear_operator [2013/09/13 21:50] nikolaj |
bounded_linear_operator [2013/09/13 21:51] nikolaj |
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| ===== Bounded linear operator ===== | ===== Bounded linear operator ===== | ||
| ==== Definition ==== | ==== Definition ==== | ||
| - | | @#88DDEE: $X,Y$ ... normed vector spaces | | + | | @#88DDEE: $V,W$ ... normed vector spaces | |
| - | | @#FFBB00: $A\mathcal{BoundedLinOp}(X,Y)$ | | + | | @#FFBB00: $A\in\mathrm{BoundedLinOp}(V,W)$ | |
| + | |||
| + | | @#55EE55: $A\in\mathrm{Hom}(V,W)$ | | ||
| | @#DDDDDD: $M\in\mathbb R, M>0$ | | | @#DDDDDD: $M\in\mathbb R, M>0$ | | ||
| - | | $x\in X$ | | + | | $v\in V$ | |
| - | | @#55EE55: $\exists M.\ \Vert Ax\Vert_Y\le M\Vert x\Vert_X $ | | + | | @#55EE55: $\exists M.\ \Vert Av\Vert_W\le M\Vert v\Vert_V $ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
