Some fundamental geometric and topological properties of generalized Orlicz‐Lorentz function spaces

Generalized Orlicz‐Lorentz function spaces Λ φ generated by Musielak‐Orlicz functions φ satisfying some growth and regularity conditions (cf. 34 and 38) are investigated. A regularity condition \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Delta ^{\Lambda }_{2}$\end{document} for φ is defined in such a way that it guarantees many positive topological and geometric properties of Λ φ . The problems of the Fatou property, order continuity (separability) and the Kadec‐Klee property with respect to the local convergence in measure of Λ φ are considered. Moreover, some embeddings between Λ φ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of Λ φ are presented. This paper generalizes the results from 20. Analogous results in the sequence case were presented in 10 and 11, but the techniques in the function case are different. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim