On the Riesz dual of \bf L¹(μ)

2020/04/01 by Arnoud van Rooij, van Rooij, Arnoud
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.2004.00745

openalex publication_date 2020/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, (X, , \A, , \μ) is a measure apace. A classical\nresult establishes a Riesz isomorphism between L¹(\μ)^\∼ and\nL^\∞(\μ) in case the measure \μ is \σ-finite. In general,\nthere still is a natural Riesz homomorphism \Φ: L^\∞(\μ) \→\nL¹(\μ)^\∼, but it may not be injective or surjective. We prove that\nalways the range of \Φ is an order dense Riesz subspace of\nL¹(\μ)^\∼. If \μ is semi-finite, then L¹(\μ)^\∼ is a\nDedekind completion of L^\∞(\μ).\n

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