论文标题
MacPhail的定理重新审视
Macphail's Theorem revisited
论文作者
论文摘要
1947年，M。S。MacPhail在$ \ ell_ {1} $中构建了一个无条件收敛但不会绝对收敛的系列。根据文献，这一结果帮助Dvoretzky和Rogers最终回答了Banach Space理论的长期存在的问题，它表明在所有无限维度Banach空间中，存在一个无条件的总结序列，这绝对是绝对可总结的。更准确地说，dvoretzky- rogers theorem断言，在每个无限二维Banach Space $ e $中都存在无条件收敛的系列$ {\ textstyle \ sum} x^{（j）} $ x^{（J）} \ vert^{^{2- \ varepsilon}} = \ infty $对于所有$ \ varepsilon> 0。适用于所有$ \ varepsilon> 0的构造。$
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 $\varepsilon>0.$
