A note on measures related to compactness and the Banach-Saks property in \(l_1\) (Q6156374)

In this note, the author provides an alternative short proof that several quantities describing non-compactness, weak non-compactness and failure of the Banach-Saks property coincide in \(\ell_1\). This result has been proven originally in [\textit{K.~Tu}, Arch. Math. 117, No.~3, 315--322 (2021; Zbl 1476.46034)]. More precisely, let: \begin{itemize} \item \(\chi\) denote the Hausdorff measure of non-compactness; \item \(\omega\) denote the De Blasi measure of weak non-compactness; \item \(\beta\) denote the separation measure of non-compactness; \item \(\gamma\) denote the convex separation measure of weak non-compactness; \item \(\varphi\) denote the arithmetic separation measure of deviation from the Banach-Saks property. \end{itemize} The result is that, for any bounded subset \(D\) of \(\ell_1\), it is true that \[ \gamma(D) = \varphi(D) = \beta(D) = 2 \chi(D) = 2 \omega(D). \] The proof relies only on the Schur property of \(\ell_1\), which trivially implies that \(\chi(D)=\omega(D)\), the known formula for \(\chi\) for subsets of \(\ell_1\), and the Bessaga-Pełczyński selection principle.