# Bounded linear map

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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