On some geometric properties of generalized Orlicz-Lorentz sequence spaces

In this paper, we continue investigations concerning generalized Orlicz-Lorentz sequence spaces lambda(phi) 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 phi satisfies condition delta(lambda)(2), then lambda(phi) has the coordinatewise Kadec-Klee property. Next, monotonicity properties are considered. In order to get sufficient conditions for uniform monotonicity of the space lambda(phi), a strong condition of delta(2) type and the notion of regularity of function phi are introduced. Finally, criteria for non-squareness of lambda(phi), of their subspaces of order continuous elements (lambda(phi))(a) as well as of finite dimensional subspaces lambda(n)(phi) of lambda(phi) are presented. C) 2012 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.