# A $\xi$-weak Grothendieck compactness principle

## TL;DR

This paper introduces a hierarchy of weak compactness concepts in Banach spaces, generalizing classical results and providing a new characterization of $\xi$-Schur spaces through $\xi$-weakly compact sets.

## Contribution

It defines $\xi$-weakly precompact and $\xi$-weakly compact sets, proves their equivalence with weak closures, and extends Grothendieck's compactness principle to this hierarchy.

## Key findings

- Characterization of $\xi$-weakly precompact sets via weak closure
- Quantified Grothendieck's compactness principle for $\xi$-weakly compact sets
- New characterization of $\xi$-Schur spaces

## Abstract

For $0\leqslant \xi\leqslant \omega_1$, we define the notion of $\xi$-weakly precompact and $\xi$-weakly compact sets in Banach spaces and prove that a set is $\xi$-weakly precompact if and only if its weak closure is $\xi$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\leqslant \xi\leqslant \omega_1$, we prove that a Banach space $X$ has the $\xi$-Schur property if and only if every $\xi$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence.