Definition:Invertible Bounded Linear Transformation/Inner Product Space

Definition
Let $\struct {V, \innerprod \cdot \cdot}$ and $\struct {U, \innerprod \cdot \cdot}$ be inner product spaces.

Let $\map \BB {V, U}$ be the space of bounded linear transformations from $V$ to $U$.

Let $A \in \map \BB {V, U}$ be a bounded linear transformation.

We say that $A$ is invertible as a bounded linear transformation :


 * $A$ has an inverse mapping that is a bounded linear transformation.

That is:


 * there exists a bounded linear transformation $B$ 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.

Also see

 * Definition:Invertible Bounded Linear Operator/Inner Product Space