Weakly convergent sequence coefficient of product space

Zhang, Guang Lu

doi:10.1090/S0002-9939-1993-1152993-1

Weakly convergent sequence coefficient of product space
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Proc. Amer. Math. Soc. 117 (1993), 637-643 Request permission

Abstract:

W. L. Bynum introduced the weakly convergent sequence coefficient $\operatorname {WCS} (X)$ of the Banach space $X$ as $\operatorname {WCS} (X) = {\text {sup}}\{ M:{\text {for each weakly convergent sequence}}\;\{ {x_n}\} \;{\text {in}}\;X,\;{\text {there is some }}y \in \overline {\operatorname {co} } (\{ {x_n}\} )\;{\text {such that }}M \cdot \lim \sup ||{x_n} - y|| \leqslant A(\{ {x_n}\} )\}$. We consider the weakly convergent sequence coefficient of the ${l_p}$-product space $Z = (\prod \nolimits _{i = 1}^n {{X_i}{)_{lp}}}$ of the finite non-Schur space ${X_1}, \ldots ,{X_n}$ and show that $\operatorname {WCS} (Z) = \min \{ \operatorname {WCS} ({X_i}):1 \leqslant i \leqslant n\}$.

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