L-Dunford-Pettis property in Banach spaces. (English) Zbl 1374.46025

Retbi, A.; El Wahbi, B.

Methods Funct. Anal. Topol. 22, No. 4, 387-392 (2016).

A norm bounded subset \(A\) of a Banach space \(X\) is called a Dunford-Pettis (DP) set if every weakly null sequence \(\{ f_n\} \subset X'\) converges to zero uniformly on \(A\).
The authors introduce and study a dual notion. A norm bounded subset \(A\) of the dual space \(X'\) is called an L-DP set if for every weakly null sequence \(\{ x_n\}\), which is a DP set in \(X\), \[ \lim\limits_{n\to \infty}\sup\limits_{f\in A}| f(x_n)| =0. \] Connections of this property with some well-known geometric properties of Banach spaces are discussed. Another related subject is the complementability in spaces of operators from \(X\) to \(l^\infty\).

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

46B50	Compactness in Banach (or normed) spaces
46B28	Spaces of operators; tensor products; approximation properties

Keywords:

Dunford-Pettis property; Dunford-Pettis set; complementability in spaces of operators