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

  • Antonio Avilés
  • Pedro TradaceteEmail author
  • Ignacio Villanueva


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 \(1\le p\le 2\).


Banach lattice Free lattice Weakly compactly generated space 

Mathematics Subject Classification

46B42 46B25 



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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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