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

Abstract

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 Fréchet differentiable norm, then (a) Orlicz–Bochner function spaces LM0(X⁎) have the Radon–Nikodym property if and only if M∈Δ2; (b) Orlicz–Bochner function spaces EN(X) are Asplund spaces if and only if M∈Δ2. (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.