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<p>We prove the following results: (i) Every absolutely weakly compact set in a Banach lattice is absolutely weakly sequentially compact. (ii) The converse of (i) holds if <i>E</i> is separable or <i>B</i><sub><i>E</i>**</sub> is absolutely weak* compact. (iii) Every absolutely weakly compact subset of a Banach lattice is contained in the closed convex hull of an absolutely weakly null sequence if and only if the Banach lattice has the positive Schur property. Examples and applications are provided.</p>

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Grothendieck’s compactness principle for the absolute weak topology

  • 期刊论文

  • 发表日期：

    2025年7月13日

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  • Geraldo Botelho,
  • José Lucas P. Luiz,
  • Vinícius C. C. Miranda

摘要

We prove the following results: (i) Every absolutely weakly compact set in a Banach lattice is absolutely weakly sequentially compact. (ii) The converse of (i) holds if E is separable or BE** is absolutely weak* compact. (iii) Every absolutely weakly compact subset of a Banach lattice is contained in the closed convex hull of an absolutely weakly null sequence if and only if the Banach lattice has the positive Schur property. Examples and applications are provided.