\begin{definition}[Definition:Invertible Bounded Linear Transformation/Inner Product Space]
Let $\struct {V, \innerprod \cdot \cdot}$ and $\struct {U, \innerprod \cdot \cdot}$ be inner product spaces.
Let $A : V \to U$ be a bounded linear transformation.
We say that $A$ is '''invertible as a bounded linear transformation''' {{iff}}:
:$A$ has an inverse mapping that is a bounded linear transformation.
That is:
:there exists a bounded linear transformation $B : U \to V$ such that: 
::$A \circ B = I_U$
::$B \circ A = I_V$
where $I_U$ and $I_V$ are the identity mappings on $U$ and $V$ respectively.
We say that $B$ is the '''inverse''' of $A$ and write $B = A^{-1}$. 
The process of finding an $A^{-1}$ given $A$ is called '''inverting'''.
\end{definition}