On subsets of the space of bounded sequences (Q2037640)

Consider the space of bounded sequences \(l_\infty\) with the ordinary norm \[\|x\|=\sup_{k\in\mathbb{N}}|x_k|\] and ordinary semiorder, where \(\mathbb{N}\) stands for the set of positive integers. {Definition 1.} A linear functional \(B\in l_\infty^*\) is called a Banach limit if \begin{itemize} \item[1.] \(B\geq 0\), i.e., \(Bx\geq0\) for \(x\geq0\); \item[2.] \(B\mathbb{I}=1\), where \(\mathbb{I}=(1, 1, \dots)\); \item[3.] \(B(Tx)=B(x)\) for all \(x\in l_\infty\), where \(T\) is the shift operator, i.e., \(T(x_1, x_2, \dots)=(x_2, x_3, \dots)\). \end{itemize} The authors denote the set of all Banach limits by \(\mathfrak{B}\). \textit{L. Sucheston} [Am. Math. Mon. 74, 308--311 (1967; Zbl 0148.12202)] established that \[q(x)\leq Bx\leq p(x)\] for any \(x\in l_\infty\) and \(B \in \mathfrak{B}\), where \[q(x)=\lim_{n\to\infty}\inf_{m\in\mathbb{N}}\frac{1}{n}\sum_{k=m+1}^{m+n}x_k \text{ and } p(x)=\lim_{n\to\infty}\sup_{m\in\mathbb{N}}\frac{1}{n}\sum_{k=m+1}^{m+n}x_k\] are called the lower and upper Sucheston functionals, respectively. In the paper under review, the authors discuss properties of the linear spans of sets defined by using the Sucheston functionals.