Dense lineability and spaceability in certain subsets of $\ell_{\infty}$

TL;DR

This paper explores the dense lineability and spaceability of specific subsets of bounded sequences in ll_, focusing on sequences with a prescribed number of accumulation points, and extends results to non-convergent sequences and other convergence notions.

Contribution

It establishes dense lineability and spaceability results for sequences with countably many accumulation points, and dense lineability without spaceability for sequences with finitely many accumulation points.

Findings

Sequences with countably many accumulation points are densely lineable in ll_.

Sequences with countably many accumulation points are spaceable in ll_.

Sequences with finitely many accumulation points are densely lineable but not spaceable.

Abstract

We investigate dense lineability and spaceability of subsets of $\ell_\infty$ with a prescribed number of accumulation points. We prove that the set of all bounded sequences with exactly countably many accumulation points is densely lineable in $\ell_\infty$, thus complementing a recent result of Papathanasiou who proved the same for the sequences with continuum many accumulation points. We also prove that these sets are spaceable. We then consider the same problems for the set of bounded non-convergent sequences with a finite number of accumulation points. We prove that such set is densely lineable in $\ell_\infty$ and that it is nevertheless not spaceable. The said problems are also studied in the setting of ideal convergence and in the space $\mathbb{R}^\omega$.