Algebraic and topological properties of some sets in ℓ₁

Taras Banakh, et al. "Algebraic and topological properties of some sets in ℓ₁." Colloquium Mathematicae 129.1 (2012): 75-85. <http://eudml.org/doc/284356>.

@article{TarasBanakh2012,
abstract = {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: () a finite union of closed intervals; () homeomorphic to the Cantor set; 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, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable. We also show that is a dense $_δ$-set in ℓ₁ and ℐ is a true $ℱ_σ$-set. Finally we show that ℐ is spaceable while is not.},
author = {Taras Banakh, Artur Bartoszewicz, Szymon Głąb, Emilia Szymonik},
journal = {Colloquium Mathematicae},
keywords = {subsums of series; achievement set of a sequence; algebrability; strong algebrability; lineability; spaceability},
language = {eng},
number = {1},
pages = {75-85},
title = {Algebraic and topological properties of some sets in ℓ₁},
url = {http://eudml.org/doc/284356},
volume = {129},
year = {2012},
}

TY - JOUR
AU - Taras Banakh
AU - Artur Bartoszewicz
AU - Szymon Głąb
AU - Emilia Szymonik
TI - Algebraic and topological properties of some sets in ℓ₁
JO - Colloquium Mathematicae
PY - 2012
VL - 129
IS - 1
SP - 75
EP - 85
AB - 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: () a finite union of closed intervals; () homeomorphic to the Cantor set; 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, and the sets of all sequences x ∈ ℓ₁∖c₀₀ such that E(x) has the property (ℐ), () and ( ), respectively. We show that ℐ and are strongly -algebrable and is -lineable. We also show that is a dense $_δ$-set in ℓ₁ and ℐ is a true $ℱ_σ$-set. Finally we show that ℐ is spaceable while is not.
LA - eng
KW - subsums of series; achievement set of a sequence; algebrability; strong algebrability; lineability; spaceability
UR - http://eudml.org/doc/284356
ER -