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\(\varepsilon\)-weakly precompact sets in Banach spaces. (English) Zbl 1491.46017

Summary: A bounded subset \(M\) of a Banach space \(X\) is said to be \(\varepsilon \)-weakly precompact, for a given \(\varepsilon \geq 0\), if every sequence \((x_n)_{n\in \mathbb N}\) in \(M\) admits a subsequence \((x_{n_k})_{k\in \mathbb N}\) such that \[ \limsup_{k\to \infty}x^*(x_{n_k})-\liminf\limits_{k\to \infty}x^*(x_{n_k})\leq\varepsilon \] for all \(x^*\in B_{X^*}\). In this paper we discuss several aspects of \(\varepsilon\)-weakly precompact sets. On the one hand, we give quantitative versions of the following known results: (a) the absolutely convex hull of a weakly precompact set is weakly precompact (Stegall), and (b) for any probability measure \(\mu \), the set of all Bochner \(\mu\)-integrable functions taking values in a weakly precompact subset of \(X\) is weakly precompact in \(L_1(\mu,X)\) (Bourgain, Maurey, Pisier). On the other hand, we introduce a Banach space property related to the one considered by K. K. Kampoukos and S. K. Mercourakis [Topology Appl. 160, No. 9, 1045–1060 (2013; Zbl 1291.46016)] when studying subspaces of strongly weakly compactly generated spaces. We say that a Banach space \(X\) has property \(\mathfrak{KM}_w\) if there is a family \(\{M_{n,p}:n,p\in\mathbb N\}\) of subsets of \(X\) such that: (i) \(M_{n,p}\) is \(1/p\)-weakly precompact for all \(n,p\in \mathbb N\), and (ii) for each weakly precompact set \(C\subseteq X\) and for each \(p\in\mathbb N\), there is \(n\in\mathbb N\) such that \(C \subseteq M_{n,p}\). All subspaces of strongly weakly precompactly generated spaces have property \(\mathfrak{KM}_w\). Among other things, we study the three-space problem and the stability under unconditional sums of property \(\mathfrak{KM}_w\).

MSC:

46B50 Compactness in Banach (or normed) spaces
46G10 Vector-valued measures and integration

Citations:

Zbl 1291.46016
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References:

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