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!



A characterization of Grothendieck spaces via convex block subsequences
Let $X$ be such that every weak\*-null sequence $(x\_n^\*)$ in $X^\*$ admits a convex block sequence which is norm-null. Since subsequences of weak\*-null sequences are again weak\*-null, by the converse to Mazur it follows that $(x\_n^\*)$ is weakly null.


