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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!

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  • $\begingroup$ I suppose you meant to say that “...$X$ is a Grothendieck space if each weak*-null (rather than weak-null) sequence in $X^*$ is weakly null.” $\endgroup$
    – Jack L.
    Nov 12 '20 at 10:33
  • $\begingroup$ Thanks, Jack. I have made those corrections. $\endgroup$ Nov 12 '20 at 13:55
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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.

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  • $\begingroup$ Thanks, Ben. But what is the converse to Mazur ? $\endgroup$ Nov 12 '20 at 23:48
  • $\begingroup$ @DongyangChen Mazur says that every subsequence of a weakly null sequence in a Banach space $X$ admits a norm-null convex block sequence. Conversely, if $(x_n)$ is not weakly null, then there are $x^*\in X^*$, $\varepsilon>0$, and a subsequence $(x_{n_k})$, such that $x^*(x_{n_k})>\varepsilon$ for all $k$. This prevents there from being any norm-null convex block sequence of $(x_{n_k})$. $\endgroup$
    – Ben W
    Nov 13 '20 at 2:21
  • $\begingroup$ You are right, Ben. $\endgroup$ Nov 13 '20 at 16:05

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