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The free Banach lattices generated by \(\ell _p\) and \(c_0\)

  • Antonio Avilés
  • Pedro TradaceteEmail author
  • Ignacio Villanueva
Article
  • 4 Downloads

Abstract

We prove that, when \(2<p<\infty \), in the free Banach lattice generated by \(\ell _p\) (respectively by \(c_0\)), the absolute values of the canonical basis form an \(\ell _r\)-sequence, where \(\frac{1}{r} = \frac{1}{2} + \frac{1}{p}\) (respectively an \(\ell _2\)-sequence). In particular, in any Banach lattice, the absolute values of any \(\ell _p\) sequence always have an upper \(\ell _r\)-estimate. Quite surprisingly, this implies that the free Banach lattices generated by the nonseparable \(\ell _p(\Gamma )\) for \(2<p<\infty \), as well as \(c_0(\Gamma )\), are weakly compactly generated whereas this is not the case for \(1\le p\le 2\).

Keywords

Banach lattice Free lattice Weakly compactly generated space 

Mathematics Subject Classification

46B42 46B25 

Notes

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Copyright information

© Universidad Complutense de Madrid 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de MurciaMurciaSpain
  2. 2.Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)Consejo Superior de Investigaciones CientíficasMadridSpain
  3. 3.Departamento de Análisis Matemático y Matemática Aplicada and Instituto de Matemática Interdisciplinar-IMIUniversidad Complutense de MadridMadridSpain

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