In this paper, we show a quantitative version of the theorem stating that relatively weakly compact sets in \(\ell _1\) coincide with those having the Banach-Saks property. Namely, we prove that the measure of the weak noncompactness based on the Eberlein double limit criterion is equal to the measure of the non-Banach-Saks property defined by the arithmetic separation of sequences.

中文翻译：

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定量弱紧集和Banach-Saks集在$$ \ ell _1 $$ℓ1中

在本文中，我们显示了定理的定量形式，指出\（\ ell _1 \）中相对较弱的紧集与具有Banach-Saks属性的紧集一致。即，我们证明了基于Eberlein双极限准则的弱非紧实度的度量等于序列的算术分离所定义的non-Banach-Saks性质的度量。