Difference between revisions of "Bounded linear map"
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Given two [[normed space|normed spaces]] {{M|(X,\Vert\cdot\Vert_X)}} and {{M|(Y,\Vert\cdot\Vert_Y)}} and a [[linear map]] {{M|L:X\rightarrow Y}}, we say that{{rAPIKM}}: | Given two [[normed space|normed spaces]] {{M|(X,\Vert\cdot\Vert_X)}} and {{M|(Y,\Vert\cdot\Vert_Y)}} and a [[linear map]] {{M|L:X\rightarrow Y}}, we say that{{rAPIKM}}: | ||
* {{M|L}} is bounded if (and only if) | * {{M|L}} is bounded if (and only if) | ||
| − | ** {{M|\exists A | + | ** {{M|\exists A\ge 0\ \forall x\in X\left[\Vert L(x)\Vert_Y\le A\Vert x\Vert_X\right]}} |
==See also== | ==See also== | ||
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Definition
Given two normed spaces [ilmath](X,\Vert\cdot\Vert_X)[/ilmath] and [ilmath](Y,\Vert\cdot\Vert_Y)[/ilmath] and a linear map [ilmath]L:X\rightarrow Y[/ilmath], we say that[1]:
- [ilmath]L[/ilmath] is bounded if (and only if)
- [ilmath]\exists A\ge 0\ \forall x\in X\left[\Vert L(x)\Vert_Y\le A\Vert x\Vert_X\right][/ilmath]
See also
- Equivalent conditions for a linear map between two normed spaces to be continuous everywhere - of which being bounded is an equivalent statement
