\begin{definition}[Definition:Bounded Linear Operator/Inner Product Space]
Let $\struct {V, \innerprod \cdot \cdot}$ be an inner product space.
Let $\norm \cdot$ be the inner product norm for $V$. 
Let $A : V \to V$ be a linear operator.
We say that $A$ is a '''bounded linear operator''' {{iff}}:
:there exists $c > 0$ such that $\norm {A v} \le c \norm v$ for each $v \in V$.
That is, a '''bounded linear operator''' on an inner product space is a bounded linear transformation from the space to itself.
\end{definition}