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# Weak Sequential Completeness of $\mathcal K(X,Y)$

Published:2012-03-05
Printed: Sep 2013
• Qingying Bu,
Department of Mathematics, University of Mississippi, University, MS 38677, USA
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## Abstract

For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are weakly sequentially complete and every weakly compact operator from $X$ to $Y$ is compact then the space of all compact operators from $X$ to $Y$ is weakly sequentially complete. The converse is also true if, in addition, either $X^\ast$ or $Y$ has the bounded compact approximation property.
 Keywords: weak sequential completeness, reflexivity, compact operator space
 MSC Classifications: 46B25 - Classical Banach spaces in the general theory 46B28 - Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]

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