Some topological and geometric properties of generalized Euler sequence space

In this paper, we introduce the Euler sequence space e ^r (p) of nonabsolute type and prove that the spaces e ^r (p) and l(p) are linearly isomorphic. Besides this, we compute the α-, β- and γ-duals of the space e ^r (p). The results proved herein are analogous to those in ALTAY, B.—BASŠAR, F.: On the paranormed Riesz sequence spaces of non-absolute type, Southeast Asian Bull. Math. 26 (2002), 701–715] for the Riesz sequence space r ^q (p). Finally, we define a modular on the Euler sequence space e ^r (p) and consider it equipped with the Luxemburg norm. We give some relationships between the modular and Luxemburg norm on this space and show that the space e ^r (p) has property (H) but it is not rotund (R).