Smoothness of the Orlicz Norm in Musielak-Orlicz Function Spaces

TL;DR

This paper characterizes support functionals and smooth points in Musielak-Orlicz spaces with the Orlicz norm, providing criteria for smoothness and explicit expressions for dual space norms.

Contribution

It offers a new characterization of smoothness and support functionals in Musielak-Orlicz spaces, advancing understanding of their geometric structure.

Findings

Criteria for smoothness of Musielak-Orlicz spaces established

Explicit expressions for dual norms derived

Support functionals characterized in the space

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 $(L_{0}^{\Phi})^{*}$, the topological dual of $L_{0}^{\Phi}$, are proved for arbitrary Musielak-Orlicz functions.