Orlicz spaces with a o–O type structure

We study non reflexive Orlicz spaces LΨ and their Morse subspace MΨ, i.e. the closure of L∞ in MΨ to determine when (MΨ, LΨ) can be described as having an o–O type structure with respect to an equivalent norm on LΨ. Examples of classes of Young functions for which the answer is affirmative are provided, but also examples are given to show that this is not possible for all non-reflexive Orlicz spaces. An equivalent expression of the distance in LΨ to MΨ, induced by the new norm, is also provided.