Definition:Bounded Linear Operator/Inner Product Space

Definition
Let $\HH$ be a Hilbert space.

Let $A: \HH \to \HH$ be a linear operator.

$A$ is a bounded linear operator


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

That is, a bounded linear operator is a bounded linear transformation from a Hilbert space to itself.