A Banach space $E$ is called *Grothendieck* if every weak\* convergent sequence in the dual space $E^\*$ is weakly convergent. A typical example of a Grothendieck space is $\ell\_\infty$. Diestel proved the following characterization of Grothendieck Banach spaces:



> 
> For every Banach space $E$, TFAE:
> 
> 
> 1. $E$ is Grothendieck,
> 2. for each separable Banach space $F$, every bounded operator $T\colon E\to F$ is weakly compact,
> 3. every bounded operator $T\colon E\to c\_0$ is weakly compact.
> 
> 
> 


**My question:** Can we replace the space $c\_0$ in item 3 by some/any other fixed separable space? In particular, can we use here $\ell\_1$?


(By "other" I of course mean "non-isomorphic to $c\_0$".)



Weakly compact operators into $c_0$ and other separable spaces
In the light of Professor Johnson's comment, we cannot replace $c\_0$ with another $X$ unless $X$ contains a copy of $c\_0$.


Indeed, let $X$ be a Banach space that contains no isomorphic copy of $c\_0$. Consider a $C(K)$ space that is not a Grothendieck space (e.g., $C([0,1])$ ). Any bounded linear $T:C(K)\to X$ is unconditionally converging since it does not fix a copy of $c\_0$, thus weakly compact since $C(K)$ has property $(V)$.


Generally, if $E$ (may not be a Grothendieck space) has property (V) & $X$ contains no copy of $c\_0$, then every $T\in B(E,X)$ is weakly compact.


