On some geometric properties of generalized Orlicz-Lorentz sequence spaces

In this paper, we continue investigations concerning generalized Orlicz-Lorentz sequence spaces λφ initiated in the papers of Foralewski et al. (2008) [16,17] (cf. also Foralewski (2011) [11,12]). As we will show in Examples 1.1-1.3 the class of generalized Orlicz-Lorentz sequence spaces is much more wider than the class of classical Orlicz-Lorentz sequence spaces. Moreover, it is shown that if a Musielak-Orlicz function φ satisfies condition δ2λ, then λφ has the coordinatewise Kadec-Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space λφ, a strong condition of δ2 type and the notion of regularity of function φ are introduced. Finally, criteria for non-squareness of λφ, of their subspaces of order continuous elements (λφ)a as well as of finite dimensional subspaces λφn of λφ are presented.