Definition:Bounded Linear Transformation

Definition
Let $\HH$ and $\KK$ be Hilbert spaces.

Let $A: \HH \to \KK$ be a linear transformation.

Then $A$ is a bounded linear transformation :


 * $\exists c > 0: \forall h \in \HH: \norm {A h}_\KK \le c \norm h_\HH$

where $\norm {\,\cdot\,}_\HH$ and $\norm {\,\cdot\,}_\KK$ denote the norms.

Also see

 * Definition:Norm on Bounded Linear Transformation
 * Definition:Space of Bounded Linear Transformations


 * Continuity of Linear Transformations: a linear transformation between Hilbert spaces is bounded it is continuous.