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ON A QUESTION OF BOURAS CONCERNING WEAK COMPACTNESS OF ALMOST DUNFORD–PETTIS SETS

  • JIN XI CHEN (a1) and LEI LI (a2)

Abstract

We give a positive answer to the question of Bouras [‘Almost Dunford–Pettis sets in Banach lattices’, Rend. Circ. Mat. Palermo (2) 62 (2013), 227–236] concerning weak compactness of almost Dunford–Pettis sets in Banach lattices. That is, every almost Dunford–Pettis set in a Banach lattice $E$ is relatively weakly compact if and only if $E$ is a $\mathit{KB}$ -space.

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[1]Aliprantis, C. D. and Burkinshaw, O., Positive Operators (reprint of the 1985 original) (Springer, Dordrecht, 2006).
[2]Bouras, K., ‘Almost Dunford–Pettis sets in Banach lattices’, Rend. Circ. Mat. Palermo (2) 62 (2013), 227236.
[3]Chen, J. X., Chen, Z. L. and Ji, G. X., ‘Almost limited sets in Banach lattices’, J. Math. Anal. Appl. 412 (2014), 547553.
[4]Meyer-Nieberg, P., Banach Lattices (Universitext) (Springer, Berlin, 1991).
[5]Wnuk, W., ‘Banach lattices with the weak Dunford–Pettis property’, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 42 (1994), 227236.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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