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تسجيل مستخدم جديدMacphails Theorem revisited
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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 ${textstylesum}x^{(j)}$ such that ${textstylesum}Vert x^{(j)}Vert^{^{2-varepsilon}}=infty$ for all $varepsilon>0.$ Their proof is non-constructive and Macphails result for $E=ell_{1}$ provides a constructive proof just for $varepsilongeq1.$ In this note we revisit Machphails paper and present two alternative constructions that work for all $varepsilon>0.$
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