Macphail's Theorem revisited

Abstract

In 1947, M. S. Macphail constructed a series in $\ell_{1}$ that converges unconditionally but does not converge absolutely. According to the literature, this result helped Dvoretzky and Rogers to finally answer a long standing problem of Banach Space Theory, by showing that in all infinite-dimensional Banach spaces, there exists an unconditionally summable sequence that fails to be absolutely summable. More precisely, the Dvoretzky--Rogers Theorem asserts that in every infinite-dimensional Banach space $E$ there exists an unconditionally convergent series ${\textstyle\sum}x^{(j)}$ such that ${\textstyle\sum}\Vert x^{(j)}\Vert^{^{2-\varepsilon}}=\infty$ for all $\varepsilon>0.$ Their proof is non-constructive and Macphail's result for $E=\ell_{1}$ provides a constructive proof just for $\varepsilon\geq1.$ In this note we revisit Machphail's paper and present two alternative constructions that work for all $\varepsilon>0.$