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

http://dx.doi.org/10.4153/CMB-2011-202-9
7 pages

  Published:2012-03-05
 

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.