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arXiv · 2404.14939

Approximative compactness in B\"ochner spaces

Guillaume Grelier·Jaime San Martín
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Abstract

For any $p\in[1,\infty)$, we prove that the set of simple functions taking at most $k$ different values is proximinal in B\"ochner spaces $L^p(X)$ whenever $X$ is a dual Banach space with $w^*$-sequentially compact unit ball. With additional properties on $X$ and its norm, we show these sets are approximatively $w^*$-compact for $p\in(1,\infty)$ and even approximatively norm-compact under stronger hypothesis.

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Guillaume Grelier, Jaime San Martín. 2024-04-23. Approximative compactness in B\"ochner spaces. https://arxiv.org/abs/2404.14939

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