Difference between revisions of "Bounded linear map"
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| − | + | {{Stub page|needs fleshing out|grade=B}} | |
| + | ==Definition== | ||
| + | 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|\exists A\ge 0\ \forall x\in X\left[\Vert L(x)\Vert_Y\le A\Vert x\Vert_X\right]}} | ||
| + | ==See also== | ||
| + | * [[Equivalent conditions for a linear map between two normed spaces to be continuous everywhere]] - of which being bounded is an equivalent statement | ||
| + | |||
| + | ==References== | ||
| + | <references/> | ||
{{Definition|Linear Algebra|Functional Analysis|Topology|Metric Space}} | {{Definition|Linear Algebra|Functional Analysis|Topology|Metric Space}} | ||
Latest revision as of 21:30, 19 April 2016
Stub grade: B
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needs fleshing out
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
