Kolmogorov\unicodex2013Riesz compactness in asymptotic L_p spaces

2025/07/20 by Nuno J. Alves, Alves, Nuno J.
Computer Science · Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2507.15102

openalex publication_date 2025/07/20 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/29

Abstract

We extend the classical Kolmogorov-Riesz compactness theorem to the setting of asymptotic L_p spaces on ℝⁿ. These are nonlocally convex F-spaces that contain the standard L_p spaces as dense subspaces and include all measurable functions supported on sets of finite measure. In contrast with the classical L_p setting, an additional almost equiboundedness condition is needed, and we prove that together with the natural tail and translation conditions it characterizes relative compactness. We conclude with illustrative examples.

Citations

• Relation between asymptotic L_p-convergence and some classical modes of convergence

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