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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] (current)nikolaj Both sides previous revision Previous revision 2016/07/31 12:42 nikolaj 2014/03/21 11:11 external edit2013/09/13 21:51 nikolaj 2013/09/13 21:50 nikolaj 2013/09/13 21:50 nikolaj 2013/09/13 21:49 nikolaj created 2016/07/31 12:42 nikolaj 2014/03/21 11:11 external edit2013/09/13 21:51 nikolaj 2013/09/13 21:50 nikolaj 2013/09/13 21:50 nikolaj 2013/09/13 21:49 nikolaj created Line 2: Line 2: ==== 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]] 