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}}:

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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

References

  1. Analysis - Part 1: Elements - Krzysztof Maurin