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<p>We give a characterization when a Banach lattice, with a discrete dual, is itself discrete. We prove that every AM Banach lattice with unit, whose topological dual is discrete, is likewise discrete. We also describe discrete elements of the topological dual of the Banach lattice of continuous functions on a compact Hausdorff space, and we give a new proof of the non-discreteness of the dual <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ell '_\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>ℓ</mi> <mi>∞</mi> <mo>′</mo> </msubsup> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\ell _\infty \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

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Discreteness and duality in Banach lattices

  • 期刊论文

  • 发表日期：

    2026年7月17日

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  • Redouane Nouira

摘要

We give a characterization when a Banach lattice, with a discrete dual, is itself discrete. We prove that every AM Banach lattice with unit, whose topological dual is discrete, is likewise discrete. We also describe discrete elements of the topological dual of the Banach lattice of continuous functions on a compact Hausdorff space, and we give a new proof of the non-discreteness of the dual \(\ell '_\infty \) ℓ ∞ ′ of \(\ell _\infty \) ℓ ∞ .