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

Generalized Orlicz-Lorentz function spaces Lambda(phi) generated by Musielak-Orlicz functions phi satisfying some growth and regularity conditions (cf. [34] and [38]) are investigated. A regularity condition Delta(Lambda)(2) for phi is defined in such a way that it guarantees many positive topological and geometric properties of Lambda(phi). The problems of the Fatou property, order continuity (separability) and the Kadec-Klee property with respect to the local convergence in measure of Lambda(phi) are considered. Moreover, some embeddings between Lambda(phi) 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 Lambda(phi) 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. (C) 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim