# Definition:Invertible Bounded Linear Transformation

*This page is about invertibility in the context of Bounded Linear Transformation. For other uses, see invertible.*

## Definition

### Normed Vector Space

Let $\struct {V, \norm \cdot_V}$ and $\struct {U, \norm \cdot_U}$ be normed vector spaces.

Let $A : V \to U$ be a bounded linear transformation.

We say that $A$ is **invertible as a bounded linear transformation** if and only if:

- $A$ 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**.

### 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** if and only if:

- $A$ 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**.

## Also see

- Definition:Inverse Element, of which this is an instantiation.