For 0 ≤ ξ ≤ ω1, we define the notion of ξ-weakly precompact and ξ-weakly compact sets in Banach spaces and prove that a set is ξ-weakly precompact if and only if its weak closure is ξ-weakly compact. We prove a quantified version of Grothendieck’s compactness principle and the characterisation of Schur spaces obtained in [7] and [9]. For 0 ≤ ξ ≤ ω1, we prove that a Banach space X has the ξ-Schur property if and only if every ξ-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence. The ξ = 0 and ξ= ω1 cases of this theorem are the theorems of Grothendieck and [7], [9], respectively.

中文翻译：

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ξ-弱格洛腾迪克紧致原理

对于 0 ≤ξ≤ω1，我们定义的概念ξ- 弱预压实和ξ- Banach 空间中的弱紧集并证明一个集合是ξ- 弱预紧当且仅当它的弱闭包是ξ-弱紧凑。我们证明了格洛腾迪克紧致性原理的量化版本以及在 [7] 和 [9] 中获得的 Schur 空间的表征。对于 0 ≤ξ≤ω1, 我们证明一个 Banach 空间X有ξ-Schur 财产当且仅当ξ- 弱紧集包含在弱空（等效地，范数空）序列的封闭凸包中。这ξ= 0 和ξ=ω1这个定理的例子分别是 Grothendieck 和 [7]、[9] 的定理。