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

Part of: Normed linear spaces and Banach spaces; Banach lattices Topological linear spaces and related structures

Published online by Cambridge University Press:  02 April 2015

JIN XI CHEN  and
LEI LI
JIN XI CHEN*
Affiliation:
Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, PR China email jinxichen@home.swjtu.edu.cn
LEI LI
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 300071, PR China email leilee@nankai.edu.cn
*
✉jinxichen@home.swjtu.edu.cn
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Abstract

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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.

Keywords

almost Dunford–Pettis setL-weakly compact setthe positive Schur propertyKB-spaceBanach lattice

MSC classification

Primary: 46B42: Banach lattices
Secondary: 46A40: Ordered topological linear spaces, vector lattices 46B50: Compactness in Banach (or normed) spaces
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 1 , August 2015 , pp. 111 - 114
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

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Bouras, K., ‘Almost Dunford–Pettis sets in Banach lattices’, Rend. Circ. Mat. Palermo (2) 62 (2013), 227–236.CrossRefGoogle Scholar
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Wnuk, W., ‘Banach lattices with the weak Dunford–Pettis property’, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 42 (1994), 227–236.Google Scholar