On Distribution and Almost Convergence of Bounded Sequences

In this paper, we give the concepts of properly distributed and simply distributed sequences, and prove that they are almost convergent. Basing on these, we review the work of Feng and Li [Feng, B. Q. and Li, J. L., Some estimations of Banach limits, J. Math. Anal. Appl. 323(2006) No. 1, 481-496. MR2262220 46B45 (46A45).], which is shown to be a special case of our generalized theory. 1. preliminary and background Let l be the Banach space of bounded sequences of real numbers x := {x(n)}n=1 with norm ‖x‖∞ = sup |x(n)|. As an application of Hahn-Banach theorem, a Banach limit L is a bounded linear functional on l, which satisfies the following properties: (a)If x := {x(n)}n=1 ∈ l ∞ and x(n) ≥ 0, then L(x) ≥ 0; (b)If x := {x(n)}n=1 ∈ l ∞ and Tx = {x(2), x(3), . . .}, then L(x) = L(Tx), where T is the translation operator ; (c)L(1) = 1, where 1 := {1, 1, . . .}; (d)‖L‖ = 1; (e)If x := {x(n)}n=1 ∈ c, then L(x) = limn→∞ x(n), where c is the Banach subspace of l consisting of convergent sequences. Since the Hahn-Banach norm-preserving extension is not unique, there must be many Banach limits in the dual space of l, and usually different Banach limits have different values at the same element in l. However, there indeed exist sequences whose values of all Banach limits are the same. Condition (e) is a trivial example. Besides that, there also exist nonconvergent sequences satisfying this property, for such examples please see [1] and [2]. In [3], G. G. Lorentz called a sequence x := {x(n)}n=1 almost convergent, if all Banach limits of x, L(x), are the same. In his paper, Lorentz proved the following criterion for almost convergent sequences: Theorem 1.1. A sequence x := {x(n)}n=1 ∈ l ∞ is almost convergent if and only if