β-Duality of the Sequence Space ℓpL and a Geometric Realization via Wulff Shapes

In the present paper, we investigate the structural and geometric properties of the non-absolute type sequence space ℓpλ. Our primary novel contribution lies in demonstrating that the natural norm on its β-dual coincides with the dual norm of ℓpλ. We then investigate geometric properties of the space and prove that ℓpλ inherits uniform convexity and uniform smoothness from the classical space ℓp for 1 p ∞, while these properties fail in the case p = 1. Finally, by considering finite-dimensional truncations, we interpret the dual norm geometrically through the Wulff construction. This yields a visualization of the norm unit sphere as a Wulff shape and shows that geometric features of the resulting shape reflect the convexity and smoothness properties of the underlying sequence space.