A note on the Banaś modulus of smoothness in the Bynum space

Recently, the Banaś modulus of smoothness for the Bynum space b 2 , ∞ was obtained by Zuo and Cui (Z. Zuo, Y. Cui, Some modulus and normal structure in Banach space, J. Inequal. Appl. 2009 (2009) 15. doi:10.1155/2009/676373. Article ID 676373). It is however not true in general. In this note, we will present the exact value for this modulus in the b 2 , ∞ space.