Bounded linear map

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Definition

Given two normed spaces $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$ and a linear map $L:X\rightarrow Y$, we say that:

• $L$ is bounded if (and only if)
  • $\exists A\ge 0\ \forall x\in X\left[\Vert L(x)\Vert_Y\le A\Vert x\Vert_X\right]$

See also

• Equivalent conditions for a linear map between two normed spaces to be continuous everywhere - of which being bounded is an equivalent statement