Smoothness in Musielak-Orlicz spaces equipped with the Orlicz norm

A formula for the distance of an arbitrary element $x$ in Musielak-Orlicz space $L^\Phi$ from the subspace $E^\Phi$ of order continuous elements is given for both (the Luxemburg and the Orlicz) norms. A formula for the norm in the dual space of $L^\Phi$ is given for any of these two norms. Criteria for smooth points and smoothness in $L^\Phi$ and $E^\Phi$ equipped with the Orlicz norm are presented.