2020/06/13 by Pellegrino, Daniel, Silva, Janiely
#15A60 #40A05 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2006.07626
In 1947, M. S. Macphail constructed a series in ℓ1 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∑x(j) such that \textstyle∑\Vert x(j)\Vert^2-ε=∞ for all ε>0. Their proof is non-constructive and Macphail's result for E=ℓ1 provides a constructive proof just for ε≥1. In this note we revisit Machphail's paper and present two alternative constructions that work for all ε>0.