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 $\sum x^{\left( j\right) }$ such that $\sum \Vert x^{(j)}\Vert ^{2-\varepsilon }=\infty$ for all $\varepsilon >0$. Their proof is non-constructive and Macphail’s result for $E=\ell _{1}$ provides a constructive proof just for $\varepsilon \ge 1$. In this note, we revisit Macphail’s paper and present two alternative constructions that work for all $\varepsilon >0.$