Smoothness of the Orlicz Norm in Musielak-Orlicz Function Spaces

Abstract

In this paper, we present a characterization of support functionals and smooth points in $L_0^{\Phi }$, the Musielak-Orlicz space equipped with the Orlicz norm. As a result, criterion for the smoothness of $L_0^{\Phi }$ is also obtained. Some expressions involving the norms of functionals in $(L_0^{\Phi }{)}^{\ast }$, the topological dual of $L_0^{\Phi }$, are proved for arbitrary Musielak-Orlicz functions.