  The three-dimensional quantum Euclidean space is an example of a
non-commutative space that is obtained from Euclidean space by $q$-deformation.
Simultaneously, angular momentum is deformed to $so_q(3)$, it acts on the
$q$-Euclidean space that becomes a $so_q(3)$-module algebra this way. In this
paper it is shown, that this algebra can be realized by differential operators
acting on $C^{\infty}$ functions on $\mathbb{R}^3$. On a factorspace of
$C^{\infty}(\mathbb{R}^3)$ a scalar product can be defined that leads to a
Hilbert space, such that the action of the differential operators is defined on
a dense set in this Hilbert space and algebraically self-adjoint becomes
self-adjoint for the linear operator in the Hilbert space. The self-adjoint
coordinates have discrete eigenvalues, the spectrum can be considered as a
$q$-lattice.
