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arXiv · 2507.15102

Kolmogorov$\unicode{x2013}$Riesz compactness in asymptotic $L_p$ spaces

Nuno J. Alves
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Abstract

We extend the classical Kolmogorov-Riesz compactness theorem to the setting of asymptotic $L_p$ spaces on $\mathbb{R}^n$. These are nonlocally convex $\mathrm{F}$-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.

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BibTeXRIS

Nuno J. Alves. 2025-07-20. Kolmogorov$\unicode{x2013}$Riesz compactness in asymptotic $L_p$ spaces. https://arxiv.org/abs/2507.15102

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