Generalized Limits and Ideal Convergence

Abstract

Let I be an ideal on N, the set of positive integers. Consider the Banach space ℓ∞ of real bounded sequences, x, with ‖x‖=supk|x_k|. A positive linear functional L on ℓ∞ is called an SI−limit if L(χK)=0 for every characteristic sequence χK of sets K⊆N for which I −limχK=0. We examine regular sublinear
functionals that both generate as well as dominate SI−limits. We also show that these results are closely related to the concept of core and multipliers for bounded sequences