Almost Summability

Abstract

In the theory of sequence spaces, an application of the well-known Hahn-Banach Extension Theorem gives rise to the notion of Banach limit which further leads to an important concept of almost convergence. That is, the lim functional defined on c can be extended to the whole of ℓ _∞ and this extended functional is known as the Banach limit [11]. In 1948, Lorentz [58] used this notion of weak limit to define a new type of convergence, known as the almost convergence. Since then a huge amount of literature has appeared concerning various generalizations, extensions, and applications of this method.