For a sequence x ∈ ℓ₁∖c₀₀, one can consider the set E(x) of all subsums of the series . 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 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 -set in ℓ₁ and ℐ is a true -set. Finally we show that ℐ is spaceable while is not.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-5,
author = {Taras Banakh and Artur Bartoszewicz and Szymon G\l \k ab and Emilia Szymonik},
title = {Algebraic and topological properties of some sets in l1},
journal = {Colloquium Mathematicae},
volume = {126},
year = {2012},
pages = {75-85},
zbl = {1267.46034},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-5}
}
Taras Banakh; Artur Bartoszewicz; Szymon Głąb; Emilia Szymonik. Algebraic and topological properties of some sets in ℓ₁. Colloquium Mathematicae, Tome 126 (2012) pp. 75-85. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm129-1-5/