On some geometric and topological properties of generalized Orlicz-Lorentz sequence spaces

Generalized Orlicz-Lorentz sequence spaces lambda(phi) generated by Musielak-Orlicz functions phi satisfying some growth and regularity conditions (see [28] and [33]) 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, the order continuity and the Kadec-Kice property with respect to the uniform convergence of the space 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), their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [ 19], [41 and [17]. (C) 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.