Approximative Compactness and Full Rotundity in Musielak-Orlicz spaces and Lorentz-Orlicz spaces

Abstract

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 {\it Studia Math.} 85 (1987), 25 -- 35. %\cite{RO}. 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 K\"othe function space which is not approximatively compact and some remark concerning the compact faces property for Musielak--Orlicz spaces are given.