Algebraic and topological properties of some sets in $\ell_1$

Taras Banakh¹ ; Artur Bartoszewicz² ; Szymon Głąb² ; Emilia Szymonik²

¹ Wydział Matematyczno-Przyrodniczy Uniwersytet Jana Kochanowskiego Świętokrzyska 15 25-406 Kielce, Poland and Department of Mathematics Ivan Franko National University of Lviv Universytetska 1 79000 Lviv, Ukraine
² Institute of Mathematics Łódź University of Technology Wólczańska 215 93-005 Łódź, Poland

Colloquium Mathematicum, Tome 129 (2012) no. 1, pp. 75-85.

Résumé

For a sequence $x \in\ell_1 \setminus c_{00}$, one can consider the set $E(x)$ of all subsums of the series $\sum_{n=1}^{\infty} x(n)$. Guthrie and Nymann proved that $E(x)$ is one of the following types of sets: $(\mathcal{I})$ a finite union of closed intervals; $(\mathcal{C})$ homeomorphic to the Cantor set; $(\mathcal{MC})$ homeomorphic to the set $T$ of subsums of $\sum_{n=1}^\infty b(n)$ where $b(2n-1) = 3/4^n$ and $b(2n) = 2/4^n$. Denote by $\mathcal I$, $\mathcal C$ and $\mathcal{MC}$ the sets of all sequences $x \in\ell_1 \setminus c_{00}$ such that $E(x)$ has the property ($\mathcal I$), ($\mathcal C$) and ($\mathcal{MC}$), respectively. We show that $\mathcal I$ and $\mathcal C$ are strongly $\mathfrak{c}$-algebrable and $\mathcal{MC}$ is $\mathfrak{c}$-lineable. We also show that $\mathcal C$ is a dense $\mathcal G_\delta$-set in $\ell_1$ and $\mathcal I$ is a true $\mathcal F_\sigma$-set. Finally we show that $\mathcal I$ is spaceable while $\mathcal C$ is not.

Détail

DOI : 10.4064/cm129-1-5

Keywords: sequence ell setminus consider set subsums series sum infty guthrie nymann proved following types sets mathcal finite union closed intervals mathcal homeomorphic cantor set mathcal homeomorphic set subsums sum infty where n denote mathcal mathcal mathcal sets sequences ell setminus has property mathcal mathcal mathcal respectively mathcal mathcal strongly mathfrak algebrable mathcal mathfrak lineable mathcal dense mathcal delta set ell mathcal mathcal sigma set finally mathcal spaceable while mathcal