On the strongest locally convex Lebesgue topology on Orlicz spaces

Abstract

Let L^φ be an Orlicz space defined by an Orlicz function φ taking only finite values with ${{\rm lim\ inf}\atop {u\rightarrow \infty}}{\varphi(u)\over u} >0$ (not necessarily convex) over a complete, σ-finite and atomless measure space and let L^φ)_n ^∼ stand for the order continuous dual of Lφ. Then the strongest locally convex Lebesgue topology τ on Lφ (= the Mackey topology τ(Lφ, (Lφ)_n ^∼) is equal to the restriction of the strongest Lebesgue topology η on $L^{\overline\varphi}$, where $\overline\varphi$ is the convex minorant of φ and τ is generated by a family of norms defined by some convex Orlicz functions.