Recall that $(y_{n})_{n}$ is a convex block subsequence of a sequence $(x_{n})_{n}$ in a Banach space $X$ provided that there exists a strictly increasing sequence of positive integers $(k_{n})_{n}$ so that $y_{n}\in \textrm{co}(x_{i})_{i=k_{n-1}+1}^{k_{n}}$ for every $n$ ($k_{0}=0$).

Let us recall that a Banach space $X$ is a Grothendieck space if each $weak^{*}$-null sequence in $X^{*}$ is weakly null. If $X$ is a Grothendieck space and if $(x^{*}_{n})_{n}$ is a $weak^{*}$-null sequence in $X^{*}$, it follows from Mazur's Theorem that $(x^{*}_{n})_{n}$ admits a convex block subsequence $(y^{*}_{n})_{n}$ that converges to $0$ in norm. I want to know whether the converse is true.

Question. Let $X$ be a Banach space so that each $weak^{*}$-null sequence in $X^{*}$ admits a convex block subsequence that converges to $0$ in norm. Is $X$ a Grothendieck space?

Thank you!

(rather thanweak*-nullweak-null) sequence in $X^*$ isweakly null.” $\endgroup$