A class of PP-convex spaces lacking normal structure

We prove that, for any β>1β>1, the space Eβ=(l2,‖⋅‖β)Eβ=(l2,‖⋅‖β) where ‖⋅‖β=max{‖⋅‖2,β‖⋅‖∞}‖⋅‖β=max{‖⋅‖2,β‖⋅‖∞} is PP-convex. It is known that, for β≥2, EβEβ lacks normal structure.