Banach limits and uniform almost summability

Abstract

Some classical Hahn–Schur Theorem-like results on the uniform convergence of unconditionally convergent series can be generalized to weakly unconditionally Cauchy series. In this paper, we obtain this type of generalization via a summability method based upon the concept of almost convergence. We also obtain a generalization of the main result in Aizpuru et al. (2003) [3] using pointwise convergence of sums indexed in natural Boolean algebras with the Vitali–Hahn–Saks Property. In order to achieve that, we first study the notion of almost convergence through its original definition (which involves Banach limits), giving a description of the extremal structure of the set of all norm-1 Hahn–Banach extensions of the limit function on c to ℓ∞. We also show the existence of norm-1 Hahn–Banach extensions of the limit function on c to ℓ∞ that are not extensions of the almost limit function and hence are not Banach limits.