On unconditionally convergent series in topological rings


  • T.O. Banakh Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine; Institute of Mathematics, Jan Kochanowski University in Kielce, 7 Uniwersytecka str., 25406, Kielce, Poland
  • A.V. Ravsky Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine https://orcid.org/0000-0003-2542-6959


topological ring, unconditional convergence, locally compact topological ring, locally compact Abelian topological group
Published online: 2022-06-30


We define a topological ring $R$ to be Hirsch, if for any unconditionally convergent series $\sum_{n\in\omega} x_i$ in $R$ and any neighborhood $U$ of the additive identity $0$ of $R$ there exists a neighborhood $V\subseteq R$ of $0$ such that $\sum_{n\in F} a_n x_n\in U$ for any finite set $F\subset\omega$ and any sequence $(a_n)_{n\in F}\in V^F$. We recognize Hirsch rings in certain known classes of topological rings. For this purpose we introduce and develop the technique of seminorms on actogroups. We prove, in particular, that a topological ring $R$ is Hirsch provided $R$ is locally compact or $R$ has a base at the zero consisting of open ideals or $R$ is a closed subring of the Banach ring $C(K)$, where $K$ is a compact Hausdorff space. This implies that the Banach ring $\ell_\infty$ and its subrings $c_0$ and $c$ are Hirsch. Applying a recent result of Banakh and Kadets, we prove that for a real number $p\ge 1$ the commutative Banach ring $\ell_p$ is Hirsch if and only if $p\le 2$. Also for any $p\in (1,\infty)$, the (noncommutative) Banach ring $L(\ell_p)$ of continuous endomorphisms of the Banach ring $\ell_p$ is not Hirsch.

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How to Cite
Banakh, T.; Ravsky, A. On Unconditionally Convergent Series in Topological Rings. Carpathian Math. Publ. 2022, 14, 266-288.