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 ∞ . We also show the existence of norm-1 HahnBanach extensions of the limit function on c to ∞ that are not extensions of the almost limit function and hence are not Banach limits.

Schur Lemma and Uniform Convergence of Series through Convergence Methods

“…Next, we will prove the main result that, in a way, mimics some ideas that appear in [5,15]. It is surprising how this result unifies all known results and it applies to most summability methods.…”

“…In Section 4, we will obtain a general Schur-type lemma for general summability methods; thus, we unify results appeared in [5,15], and finally we close the paper with a brief section with concluding remarks and open questions.…”

“…A good source of problems consists in considering a result on convergence of series which is true for the usual convergence and, to try to prove it, replacing the usual convergence by other convergence methods [5][6][7][8][9]. In this way, it is possible to see a classical result from a new point of view.…”

“…We continued the research line started in [10,11], and we aim to unify different versions of Swartz's result [5,7,15]. For instance, Schur type results were obtained for any regular matrix summability method [15] and for the Banach-Lorentz convergence [5].…”

Extremal properties of the set of vector-valued Banach limits

“…The set of all almost convergent sequences in X is usually denoted by ac .X /. According to [6,7,11,16,17] we have the following: -ac .X / is a vector subspace of`1 .X /.…”