Linear operators and linear functionals

Let ${\cal{V}}_1$ and ${\cal{V}}_2$ be vectors spaces. A mapping $\hat{A}: {\cal{V}}_1 \to {\cal{V}}_2$ is a linear operator if

$\displaystyle \hat{A}(v + w) = \hat{A}v + \hat{A}w,\quad \hat{A}(av)= a \hat{A}v,$

for all $v, w \in {\cal{V}}_1$ and $a\in F$ . ${\cal{V}}_1$ is called the domain of $\hat{A}$ and ${\cal{V}}_2$ its codomain or image. If the codomain of a linear operator is a scalar field, the operator is called a linear functional on ${\cal{V}}_1$ . The set of all linear functionals on ${\cal{V}}_1$ is called the dual space of ${\cal{V}}_1$ .

The adjoint of an operator $\hat{A}: {\cal{V}}_1 \to {\cal{V}}_2$ is the unique operator $\hat{A}^\ast$ satisfying that

$\displaystyle \langle \hat{A} g_1, g_2\rangle = \langle g_1, \hat{A}^\ast g_2\rangle .$

If $\hat{A}^\ast= \hat{A}$ the operator is self-adjoint or Hermitian

An operator $\hat{A}: {\cal{V}}_1 \to {\cal{V}}_2$ has an inverse if there exists $\hat{A}^{-1} : {\cal{V}}_2 \to {\cal{V}}_1$ such that

$\displaystyle \hat{A}^{-1} \hat A =\hat{I}_{{\cal{V}}_1}$ and $\displaystyle \quad \hat A \hat{A}^{-1} =\hat{I}_{{\cal{V}}_2},$

where $\hat{I}_{{\cal{V}}_1}$ and $\hat{I}_{{\cal{V}}_2}$ denote the identity operators in ${{\cal{V}}_1}$ and ${{\cal{V}}_2}$ , respectively. By a generalised inverse we shall mean an operator $\hat{A}^{\dagger}$ satisfying the following conditions

$\displaystyle \hat{A} \hat{A}^{\dagger} \hat{A}$	$\displaystyle =$	$\displaystyle \hat{A}$
$\displaystyle \hat{A}^{\dagger} \hat{A} \hat{A}^{\dagger}$	$\displaystyle =$	$\displaystyle \hat{A}^{\dagger}.$

The unique generalized inverse satisfying

$\displaystyle (\hat{A} \hat{A}^{\dagger})^\ast$	$\displaystyle =$	$\displaystyle \hat{A} \hat{A}^{\dagger}$
$\displaystyle (\hat{A}^{\dagger} \hat{A})^\ast$	$\displaystyle =$	$\displaystyle \hat{A}^{\dagger} \hat{A}.$

is known as the Moore Penrose pseudoinverse.

Highly Nonlinear Approximations for Sparse Signal Representation

Linear operators and linear functionals