The subspace $c_0$ is not complemented in $ac_0$

TL;DR

This paper demonstrates that the subspace c_0 of sequences converging to zero is not complemented in the space ac_0 of almost converging sequences, using an approach that extends to related inclusion chains.

Contribution

It establishes the non-complementability of c_0 in ac_0 and applies the method to the inclusion chain c_0 ⊂ A_0 ⊂ ℓ_∞.

Findings

c_0 is not complemented in ac_0

Method extends to inclusion chain c_0 ⊂ A_0 ⊂ ℓ_∞

Provides new insight into the structure of sequence spaces

Abstract

We prove that the subspace $c_0$ of sequences that converge to zero is not complemented in the space $ac_0$ of sequences that almost converge to zero. We proceed with applying the same approach to inclusion chain $c_0\subset A_0 \subset \ell_\infty$.