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

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 $$l^\infty$$.

### MSC:

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