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



Generalized Orlicz-Lorentz function spaces Λφ generated by Musielak-Orlicz functions φ satisfying some growth and regularity conditions (cf. 34 and 38) are investigated. A regularity condition equation image for φ is defined in such a way that it guarantees many positive topological and geometric properties of Λφ. The problems of the Fatou property, order continuity (separability) and the Kadec-Klee property with respect to the local convergence in measure of Λφ are considered. Moreover, some embeddings between Λφ 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 Λφ 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