Difference between revisions of "Bounded linear map"
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==Definition== | ==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}}: | 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== | ||
Latest revision as of 21:30, 19 April 2016
Stub grade: B
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needs fleshing out
Definition
Given two normed spaces (X,∥⋅∥X) and (Y,∥⋅∥Y) and a linear map L:X→Y, we say that[1]:
- L is bounded if (and only if)
- ∃A≥0 ∀x∈X[∥L(x)∥Y≤A∥x∥X]
See also
- Equivalent conditions for a linear map between two normed spaces to be continuous everywhere - of which being bounded is an equivalent statement
