Macphail's theorem revisited (Q2052370)

For most classical infinite dimensional Banach spaces, it is an easy exercise to explicitly construct a series that is unconditionally convergent, but not absolutely convergent. A notable exception is the sequence space \(\ell_1\); there, such a series was first constructed by \textit{M.~S. Macphail} [Bull. Am. Math. Soc. 53, 121--123 (1947; Zbl 0032.35702)] using a Rademacher-like system of elements of \(\ell_1\). Shortly afterwards, the Dvoretzky-Rogers theorem was proved to the effect that in every Banach space there is, given \((\lambda_n)\in \ell_2\), an unconditionally convergent series \(\sum x_n\) such that \(\|x_n\|=|\lambda_n|\) [\textit{A.~Dvoretzky} and \textit{C.~A. Rogers}, Proc. Natl. Acad. Sci. USA 36, 192--197 (1950; Zbl 0036.36303)]. However, their method of proof was not constructive. The present paper is devoted to giving explicit examples of series \(\sum x_n\) in \(\ell_p\), \(1\le p\le 2\), that are unconditionally convergent, but satisfy \(\sum \|x_n\|_p^r=\infty \) for each \(r<2\). The first of these examples only works for complex scalars, and a second example works in both the real and the complex case; the latter employs the Walsh system.