Advances in almost convergence (Q455450)

A Banach limit is a functional \(L\) on the space \(\ell^{\infty}\) of bounded real-valued sequences which is positive, left-shift-invariant and satisfies \(L(1,1,\dots)=1\). The existence of Banach limits is a well-known consequence of the Hahn-Banach extension theorem. A sequence \(x\in \ell^{\infty}\) is said to be almost convergent to the number \(a\in \mathbb{R}\) if \(L(x)=a\) for every Banach limit \(L\). This generalisation of usual convergence was introduced by \textit{G. G. Lorentz} [Acta Math., Uppsala 80, 167--190 (1948; Zbl 0031.29501)], who also proved that almost convergence is equivalent to uniform Cesàro convergence, i.e., \(x\) is almost convergent to \(a\) if and only if \[ \lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^{n}x(i+j)=a \;\;\text{uniformly\;in} \;j\in\mathbb{N}\cup\{0\}. In the first part of the paper under review, the author introduces the notion of a properly distributed sequence \(x\in \ell^{\infty}\) and proves that all properly distributed sequences are almost convergent and their generalised limit can be obtained (in theory) by approximating them via so-called simply distributed sequences. This is related to the work of \textit{B. Q. Feng} and \textit{J. L. Li} [J. Math. Anal. Appl. 323, No. 1, 481--496 (2006; Zbl 1113.46015)]. In the second part, the author considers an arbitrary (real or complex) normed space \(V\) and defines a bounded nontrivial linear functional \(L\) on the space \(\ell^{\infty}(V)\) of bounded sequences in \(V\) to be a Banach limit functional if it is left-shift-invariant and satisfies \(\|L\|\leq 1\). The existence of Banach limit functionals is proved using the Hahn-Banach extension theorem and a sequence \(x\in \ell^{\infty}(V)\) is defined to be strongly almost convergent to \(v\in V\) if \(L(x)=L(v,v,\dots)\) for every Banach limit functional \(L\). This is shown to be equivalent to uniform Cesàro convergence with respect to the norm of \(V\).