How large is the space of almost convergent sequences?

TL;DR

This paper investigates the size and structure of the space of almost convergent sequences within larger sequence spaces, using concepts like porosity, algebrability, and measure to quantify their 'largeness'.

Contribution

It provides a detailed analysis of the size and algebraic structure of almost convergent sequences compared to related sequence spaces.

Findings

The space of almost convergent sequences is large in the space of all bounded sequences.

Porosity and measure-theoretic properties are used to quantify the size of these spaces.

The algebraic structure of these spaces is characterized in terms of algebrability.

Abstract

We consider the subspaces $c$, $\widehat{c}$, $S$ of $\ell^\infty$, where $\widehat{c}$ consists of almost convergent sequences, and $S$ consists of sequences whose arithmetic means of consecutive terms are convergent. We know that $c\subset \widehat{c} \subset S$. We examine the largeness of $c$ in $\widehat{c}$, $\widehat{c}$ in $S$ and $S$ in $\ell^\infty$. We will do it from the viewpoints of porosity, algebrability and measure.