Macphail's Theorem revisited

TL;DR

This paper revisits Macphail's 1947 construction of a series in  that converges unconditionally but not absolutely, providing new constructive methods applicable for all >0, thereby enriching the understanding of Banach space theory.

Contribution

The authors present two new constructive constructions of unconditionally convergent but not absolutely convergent series in  for all >0, extending Macphail's original result.

Findings

Constructive methods for all >0

Alternative proofs of Macphail's theorem

Enhanced understanding of Banach space series convergence

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 $\varepsilon>0.$