# Highly Nonlinear Approximations for Sparse Signal Representation

## Linear operators and linear functionals

Let
and
be vectors spaces.
A mapping
is a
**linear operator** if

**domain**of and its

**codomain**or

**image**. If the codomain of a linear operator is a scalar field, the operator is called a

**linear functional**on . The set of all linear functionals on is called the

**dual space**of .

The **adjoint** of an operator
is the unique operator
satisfying that

**self-adjoint**or

**Hermitian**

An operator
has an **inverse** if there exists
such that

and

where
and
denote the identity operators in
and
,
respectively.
By a **generalised inverse**we shall mean an operator satisfying the following conditions

The unique generalized inverse satisfying

is known as the

*Moore Penrose pseudoinverse*.