# A characterization of Grothendieck spaces via convex block subsequences

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!

• 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.” Nov 12, 2020 at 10:33
• Thanks, Jack. I have made those corrections. Nov 12, 2020 at 13:55

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.
• @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 $(x_{n_k})$. Nov 13, 2020 at 2:21