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

Generalized Orlicz–Lorentz sequence spaces λ φ generated by Musielak‐Orlicz functions φ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ λ 2 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, the order continuity and the Kadec–Klee property with respect to the uniform convergence of the space 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 , their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)