The strong and uniform ball-covering properties
In this paper, we first introduce the formal definitions of the strong ball-covering property (SBCP) and the uniform ball-covering property (UBCP) for Banach spaces which implicitly appeared in the papers of Cheng, Shi and Zhang  and Fonf and Zanco . While the SBCP and the BCP are equivalent under renormings, we show that there exist Banach spaces with the BCP but failing the SBCP. We also construct a Banach space having the SBCP but without the UBCP. Then we prove that the SBCP and the UBCP for a Banach space X pass to ℓp(X) 1≤p≤∞ and Lp([0,1],X) (1≤p<∞). Moreover, if E is a Banach space with an 1-unconditional basis (en), then Banach space X has the UBCP if and only if E(X) has the UBCP, where E(X) is the Banach space of sequences (xn)⊂X with ∑n‖xn‖en converging in E and ‖(xn)‖=‖∑n‖xn‖en‖.
Journal of Mathematical Analysis and Applications
Luo, Z., & Zheng, B. (2021). The strong and uniform ball-covering properties. Journal of Mathematical Analysis and Applications, 499 (1) https://doi.org/10.1016/j.jmaa.2021.125034