The hereditary Dunford-Pettis property for \(\ell_ 1(E)\). (English) Zbl 0833.46007
Summary: A Banach space \(E\) is said to be hereditarily Dunford-Pettis if all of its closed subspaces have the Dunford-Pettis property. In this note we prove that the Banach space \(\ell_1 (E)\), of all absolutely summing sequences in \(E\) with the usual norm, is hereditarily Dunford-Pettis if and only if \(E\) is also.
MSC:
| 46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |
| 46B20 | Geometry and structure of normed linear spaces |
| 46B25 | Classical Banach spaces in the general theory |
| 46E40 | Spaces of vector- and operator-valued functions |
| 46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |
