A new type of convergence

Abstract

In his important paper (1), Lorentz defined the space f of almost convergent sequences, using the idea of Banach limits. If x ∈ l∞(R), the space of bounded real sequences, andwhere the inf is taken over all sets n(1), n(2), …, n(r) of natural numbers, then a Banach limit L may be defined as a linear functional on l∞(R) which satisfies