$\varepsilon $-weakly precompact sets in Banach spaces
Volume 262 / 2022
Abstract
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 $$ \limsup _{k\to \infty }x^*(x_{n_k})-\liminf _{k\to \infty }x^*(x_{n_k}) \leq \varepsilon $$ 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 Kampoukos and Mercourakis 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 $\mathfrak {KM}_w$.
