# A characterization of reflexivity of Banach spaces via convex block sequences

By James's Theorem, A. Ulger (Weak compactness in $$L^{1}(\mu.X)$$, Proc. Amer. Math. Soc. 113(1991),143-149.) proved that a bounded subset $$A$$ of a Banach space $$X$$ is relatively weakly compact if and only if given any sequence $$(x_{n})_{n}$$ in $$A$$, there exists a sequence $$(z_{n})_{n}$$ with $$z_{n}\in conv(x_{i}:i\geq n)$$ that converges weakly. J. Diestel, W. M. Ruess and W. Schachermayer (Weak compactness $$L^{1}(\mu,X)$$, Proc. Amer. Math. Soc. 118(1993),447-453) proved that a bounded subset $$A$$ of a Banach space $$X$$ is relatively weakly compact if and only if given any sequence $$(x_{n})_{n}$$ in $$A$$, there exists a sequence $$(z_{n})_{n}$$ with $$z_{n}\in conv(x_{i}:i\geq n)$$ that is norm convergent.

Let $$(x_{n})_{n}$$ be a sequence in a Banach space $$X$$. We say that a sequence $$(z_{n})_{n}$$ in $$X$$ is a convex block subsequence of $$(x_{n})_{n}$$ if there exists $$0=k_{0} so that $$z_{n}\in conv(x_{i})_{i=k_{n-1}+1}^{k_{n}}$$ for all $$n$$. The collection of all convex block subsequences of $$(x_{n})_{n}$$ is denoted by $$cbs((x_{n})_{n})$$. By Mazur's theorem, we get the following result:

Theorem. The following statements are equivalent for a bounded subset $$A$$ of a Banach space $$X$$

(1)$$A$$ is relatively weakly compact.

(2)Every sequence in $$A$$ admits a convex block subsequence that is norm convergent.

(3)Every sequence in $$A$$ admits a convex block subsequence that is weakly convergent.

For a bounded sequence $$(x_{n})_{n}$$ in a Banach space $$X$$. We set $$ca((x_{n})_{n})=\inf_{n}\sup_{k,l\geq n}\|x_{k}-x_{l}\|.$$ Then $$ca((x_{n})_{n})=0$$ if and only if $$(x_{n})_{n}$$ is norm convergent.

For a Banach space $$X$$, we set $$R(X)=\sup_{(x_{n})_{n}\subseteq B_{X}}\inf_{(z_{n})_{n}\in cbs((x_{n})_{n})}ca((z_{n})_{n}).$$ It follows from Theorem that $$R(X)=0$$ if $$X$$ is reflexive. But I do not know whether the converse is true.

Thank you!

Beanland/Freeman proved that an operator $$T\in\mathcal{L}(X,Y)$$ is weakly compact if and only if for every normalized basic sequence $$(x_n)\in\mathcal{NB}_X$$, the image sequence $$(Tx_n)$$ fails to dominate the summing basis $$(s_n)$$ for $$c_0$$. Consequently, by considering the identity operator, $$X$$ is reflexive if and only if none of its normalized basic sequences dominates $$(s_n)$$.
Now select $$(x_n)\in\mathcal{NB}_X$$ and $$\varepsilon>0$$. Due to $$R(X)=0$$, there is $$(z_n)\in\text{cbs}(x_n)$$ such that $$\text{ca}(z_n)<\varepsilon/2$$. Hence, there are $$k,l\in\mathbb{N}$$ such that $$\|z_k-z_l\|<\varepsilon$$. On the other hand, if $$u_k$$ and $$u_l$$ are the corresponding convex blocks of $$(s_n)$$ then $$\|u_k-u_l\|\geqslant 1$$.