Algebraic and topological properties of some sets in ℓ₁

• Autorzy: Taras Banakh , Artur Bartoszewicz , Szymon Głąb , Emilia Szymonik
• Języki publikacji: EN

Abstrakty

EN

For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series $∑_{n=1}^{∞} x(n)$. Guthrie and Nymann proved that E(x) is one of the following types of sets: (𝓘) a finite union of closed intervals; (𝓒) homeomorphic to the Cantor set; 𝓜 𝓒 homeomorphic to the set T of subsums of $∑_{n=1}^{∞} b(n)$ where b(2n-1) = 3/4ⁿ and b(2n) = 2/4ⁿ. Denote by ℐ, 𝓒 and 𝓜 𝓒 the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), (𝓒) and (𝓜 𝓒), respectively. We show that ℐ and 𝓒 are strongly 𝔠-algebrable and 𝓜 𝓒 is 𝔠-lineable. We also show that 𝓒 is a dense $𝒢_δ$-set in ℓ₁ and ℐ is a true $ℱ_σ$-set. Finally we show that ℐ is spaceable while 𝓒 is not.

Kategorie tematyczne

• 40A05: Convergence and divergence of series and sequences
• 15A03: Vector spaces, linear dependence, rank

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2012
• Tom: 129
• Numer: 1
• Strony: 75-85

Daty

wydano

2012

Twórcy

autor

Taras Banakh

• Wydział Matematyczno-Przyrodniczy, Uniwersytet Jana Kochanowskiego, Świętokrzyska 15, 25-406 Kielce, Poland
• Department of Mathematics, Ivan Franko National University of Lviv, Universytetska 1, 79000 Lviv, Ukraine

autor

Artur Bartoszewicz

• Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 93-005 Łódź, Poland

autor

Szymon Głąb

• Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 93-005 Łódź, Poland

autor

Emilia Szymonik

• Institute of Mathematics, Łódź University of Technology, Wólczańska 215, 93-005 Łódź, Poland

Identyfikatory

DOI

10.4064/cm129-1-5