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# Dunford--Pettis Properties and Spaces of Operators

Published:2009-06-01
Printed: Jun 2009
• Ioana Ghenciu
• Paul Lewis
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## Abstract

J. Elton used an application of Ramsey theory to show that if $X$ is an infinite dimensional Banach space, then $c_0$ embeds in $X$, $\ell_1$ embeds in $X$, or there is a subspace of $X$ that fails to have the Dunford--Pettis property. Bessaga and Pelczynski showed that if $c_0$ embeds in $X^*$, then $\ell_\infty$ embeds in $X^*$. Emmanuele and John showed that if $c_0$ embeds in $K(X,Y)$, then $K(X,Y)$ is not complemented in $L(X,Y)$. Classical results from Schauder basis theory are used in a study of Dunford--Pettis sets and strong Dunford--Pettis sets to extend each of the preceding theorems. The space $L_{w^*}(X^* , Y)$ of $w^*-w$ continuous operators is also studied.
 Keywords: Dunford--Pettis property, Dunford--Pettis set, basic sequence, complemented spaces of operators
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 46B28 - Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]

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