  The Malliavin derivative for a L\'evy process $(X_t)$ can be defined on the
space $\DD_{1,2}$ using a chaos expansion or in the case of a pure jump process
also via an increment quotient operator \cite{sole-utzet-vives}. In this paper
we define the Malliavin derivative operator $\D$ on the class $\mathcal{S}$ of
smooth random variables $f(X_{t_1}, ..., X_{t_n}),$ where $f$ is a smooth
function with compact support. We show that the closure of $L_2(\Om) \supseteq
\mathcal{S} \stackrel{\D}{\to} L_2(\m\otimes \mass)$ yields to the space
$\DD_{1,2}.$ As an application we conclude that Lipschitz functions map from
$\DD_{1,2}$ into $\DD_{1,2}.$
