Approximative compactness and Asplund property in Banach function spaces and in Orlicz-Bochner spaces in particular with application

In this paper it is shown that: (1) If every weak* hyperplane of X* is approximatively compact, then (a) X is an Asplund space; (b) X* has the Radon Nikodym property. (2) Criteria for approximative compactness of every weakly* hyperplane of Orlicz Bochner function spaces equipped with the Orlicz norm are given. (3) If X has a Frechet differentiable norm, then (a) Orlicz Bochner function spaces L-M(0) (X*) have the Radon Nikodym property if and only if M is an element of Delta(2); (b) Orlicz Bochner function spaces E-N (X) are Asplund spaces if and only if M is an element of Delta(2). (4) We give an important application of approximative compactness to the theory of generalized inverses for operators between Banach spaces and Orlicz Bochner function spaces. (C) 2014 Elsevier Inc. All rights reserved.