Weakly Convergent Sequence Coefficient of Product Space

W. L. Bynum introduced the weakly convergent sequence coefficient WCS(A') of the Banach space X as WCS(Jf) = sup{M: for each weakly convergent sequence {xn} in X , there is some y e co({x«}) such that M • lim sup ||x,, y\\ < A({x„})} . We consider the weakly convergent sequence coefficient of the /^-product space Z = (fTjLi Xi)t of tne finite non-Schur space Xx, ... , X„ and show that WCS(Z) = min{WCS(X,): 1 < i < n} . 0. Introduction Let X be a Banach space and {xn} a sequence of X. For x £ X, set r(x, {x„}) = limsup„ ||x„ x||. A({x„}) = lim„sup{||x, Xj\\: /', j > 77} and r({xn}) = inf{r(x, {x„}): x e co({x„})} are called the asymptotic diameter of {x„} and the Chebyshev radius of {x„} relative to co({x„}), respectively, where co({x„}) denotes the closed convex hull of {x„}. If A({x„}) lim„inf{||x, x;||: /', j > n, i / 7'}, then the sequence {x„} is called the asymptotic equidistant sequence. In order to study the normal structure of Banach spaces, Bynum [1] introduced the weakly convergent sequence coefficient WCS(X) for a reflexive Banach space X as follows: WCS(X) = sup < Af: for each weakly convergent sequence {x„}, (1) there is some y £ co({x„}) such that M • limsup||x„ y\\ < A({x„}) > . It is easy to prove that (see [2]) WCS(X) = inf{A({xn})/r({xn}): {x„} a weakly but not strongly convergent sequence in X}. But, because of the indefiniteness of y in expression (1) and the existence of r({xn}) in the denominator of expression (2), it is inconvenient to apply and Received by the editors January 2, 1990. 1991 Mathematics Subject Classification. Primary 46B20.