There is probably an embarrassingly simple counter-example to my question but I couldn't figure it out myself. Let me give it a try here.


A Banach space $X$ is *Grothendieck* if weak\*-convergent sequences in $X^\*$ converge weakly (that is, with respect to the weak topology introduced by functionals in $X^{\*\*}$). Standard examples of such spaces include reflexive spaces and $C(K)$-spaces for $K$ Stonian.


I would like to relax this condition a bit, so my question is:



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> Let $X$ be a Banach space such that $X^\*$ is weak\*-separable and let $T$ be a total subspace of $X^\*$. Suppose that each sequence $(f\_n)\_{n=1}^\infty \subset T$ which converges weak\*, converges also weakly. Can we conclude that $X$ is Grothendieck?
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Grothendieck spaces and total subspaces of the dual
Here is a natural way to build a counterexample. Let $T$ be a weakly sequentially complete space and set $X=T^\*$. Consider $T$ as a subspace of $X^\*$, so that a sequence in $T$ is weak$^\*$ convergent iff it is weakly Cauchy and hence weakly convergent to an element of $T$. You want $T$ non reflexive and $X$ should not be Grothendieck. $T=\ell\_1$ satisfies the first condition but not the second. What about $T=(\sum E\_n)\_1$ for a carefully chosen sequence $(E\_n)$ of finite dimensional spaces? You need to choose the sequence so that $X$ is not Grothendieck. One way of doing that is to make sure that $X^\*$ has a complemented non reflexive separable subspace. Now if you take a Banach space $Y$ and a sequence $E\_1\subset E\_2 \subset \dots$ with $\cup E\_n$ dense in $Y$ and set $T=(\sum E\_n)\_1$, then $Y^\*$ is isometrically isomorphic to a norm one complemented subspace of $X=T^\*$ [J] and hence $Y^{\*\*}$ is isometrically isomorphic to a norm one complemented subspace of $X^\*$. So you just need $Y$ to be non reflexive and complemented in its bidual; e.g., $Y$ can be any separable non reflexive quasi-reflexive space.


[J] Johnson, William B. A complementably [sic] universal conjugate Banach space and its relation to the approximation problem. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972). Israel J. Math. 13 (1972), 301–310 (1973). 


