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Abstract

A Banach space $X$ has property $(E)$ if every operator from $X$ into $c_0$ extends to an operator from $X^{\ast \ast }$ into $c_0$; $X$ has property $(L)$ if whenever $K\subseteq X$ is limited in $X^{\ast \ast }$, then $K$ is limited in $X$; $X$ has property $(G)$ if whenever $K\subseteq X$ is Grothendieck in $X^{\ast \ast }$, then $K$ is Grothendieck in $X$. In all of these, we consider $X$ as canonically embedded in $X^{\ast \ast }$. We study these properties in connection with other geometric properties, such as the Phillips properties, the Gelfand–Phillips and weak Gelfand–Phillips properties, and the property of being a Grothendieck space.