  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.

Functional Analysis