On rearrangement theorems in Banach spaces
A well-known consequence of the famous Steinitz lemma says that, if a subsequence of partial sums of a series \(\sum x_n\) in \(\mathbb R^n\) converges to \(x\) and \(x_n \to 0\) as \(n \to \infty\), then, in some rearrangement of terms, the series converges to \(x\). The paper under review demonstrates that in every infinite-dimensional Banach space \(X\) this result fails drastically. Namely, it is shown that \(X\) contains a series \(\sum x_n\) with the following properties: (a) some subsequence of its partial sums converges in \(X\) and all its partial sums \(\sum_{k=1}^n x_k\) are uniformly bounded; (b) \(\sum_{k=1}^\infty\|x_k\|^p<\infty\) for every \(p\in(2,+\infty)\); (c) for any permutation \(\pi:\mathbb{N}\to\mathbb{N}\) and any sequence \((\theta_n)_{n\in\mathbb{N}}\) with \(\theta_n\in\{-1,1\}\), \(n=1,2,\dots\), the series \(\sum_{k=1}^\infty\theta_kx_{\pi(k)}\) diverges in \(X\). Remark that in some spaces (for example, in \(L_2\)) the condition (b) cannot be substituted with \(\sum_{k=1}^\infty\|x_k\|^2<\infty\) [\textit{M. I. Kadets}, Usp. Mat. Nauk 9, No. 1(59), 107--109 (1954; Zbl 0055.29003)].
scientific article; zbMATH DE number 6305829
scientific article