Macphail’s theorem revisited View Full Text


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Article Info

DATE

2021-10-27

AUTHORS

Daniel Pellegrino, Janiely Silva

ABSTRACT

In 1947, M.S. Macphail constructed a series in ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}$$\end{document} 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 ∑xj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum x^{\left( j\right) }$$\end{document} such that ∑‖x(j)‖2-ε=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum \Vert x^{(j)}\Vert ^{2-\varepsilon }=\infty $$\end{document} for all ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document}. Their proof is non-constructive and Macphail’s result for E=ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E=\ell _{1}$$\end{document} provides a constructive proof just for ε≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \ge 1$$\end{document}. In this note, we revisit Macphail’s paper and present two alternative constructions that work for all ε>0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0.$$\end{document} More... »

PAGES

647-656

References to SciGraph publications

  • 2019-04-05. On the Maurey–Pisier and Dvoretzky–Rogers Theorems in BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, NEW SERIES
  • 2019-09-06. On coincidence results for summing multilinear operators: interpolation, ℓ1-spaces and cotype in COLLECTANEA MATHEMATICA
  • 1968-07. On the projection and macphail constants oflnp spaces in ISRAEL JOURNAL OF MATHEMATICS
  • 2014-06-20. Unconditional Convergence of Functional Series in Problems of Probability Theory in JOURNAL OF MATHEMATICAL SCIENCES
  • 1964-12. A proof of the Dvoretzky-Rogers theorem in ISRAEL JOURNAL OF MATHEMATICS
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    http://scigraph.springernature.com/pub.10.1007/s00013-021-01676-z

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    http://dx.doi.org/10.1007/s00013-021-01676-z

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