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

Tom 129 / 2012

Taras Banakh, Artur Bartoszewicz, Szymon Głąb, Emilia Szymonik Colloquium Mathematicum 129 (2012), 75-85 MSC: Primary 40A05; Secondary 15A03. DOI: 10.4064/cm129-1-5

Streszczenie

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.

Autorzy

  • Taras BanakhWydział 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
    e-mail
  • Artur BartoszewiczInstitute of Mathematics
    Łódź University of Technology
    Wólczańska 215
    93-005 Łódź, Poland
    e-mail
  • Szymon GłąbInstitute of Mathematics
    Łódź University of Technology
    Wólczańska 215
    93-005 Łódź, Poland
    e-mail
  • Emilia SzymonikInstitute of Mathematics
    Łódź University of Technology
    Wólczańska 215
    93-005 Łódź, Poland
    e-mail

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