Approximative compactness and full rotundity in Musielak–Orlicz spaces and Lorentz–Orlicz spaces

We prove that approximative compactness of a Banach space X is equivalent to the conjunction of reflexivity and the Kadec-Klee property of X. This means that approximative compactness coincides with the drop property defined by Rolewicz in Studia Math. 85 (1987), 25 – 35. Using this general result we find criteria for approximative compactness in the class of Musielak–Orlicz function and sequence spaces for both (the Luxemburg norm and the Amemiya norm) as well as critria for this property in the class of Lorentz–Orlicz spaces. Criteria for full rotundity of Musielak–Orlicz spaces are also presented in the case of the Luxemburg norm. An example of a reflexive strictly convex Köthe function space which is not approximatively compact and some remark concerning the compact faces property for Musielak–Orlicz spaces are given.