Dense Lineability and algebrability of $\ell^{\infty}\setminus c_0$

TL;DR

This paper demonstrates that the set of bounded sequences not converging to zero is densely lineable and algebrable, providing a positive answer to a previously posed question and extending existing results in the field.

Contribution

It establishes the dense-lineability and dense-algebrability of al^{b} extbackslash c_0, addressing an open question and expanding the understanding of algebraic structures within sequence spaces.

Findings

The set al^{b} extbackslash c_0 is dense-lineable.

The set al^{b} extbackslash c_0 is dense-algebrable.

The results extend previous work by confirming algebraic richness in sequence spaces.

Abstract

We show that the set $\ell^{\infty}\setminus c_0$ is dense-lineable and dense-algebrable answering a question posed by Nestoridis and complementing a result by Garc\'ia-Pacheco, Mart\'in and Seoane-Sep\'ulveda.