Nearly Uniform Convexity of Infinite Direct Sums

Those kinds of infinite direct sums are characterized for which NUC (NUS, respectively) is inherited from the component spaces to the direct sum. 1. NUC and the lower KK-modulus. The Banach space X is said to be nearly uniformly convex, abbreviated NUC (cf. [2]), if: for every ε > 0, there exists δ > 0 such that the convex hull conv{xn} of every sequence {xn} in the unit ball BX of X with separation sep{xn} = inf n6=m ‖xn − xm‖ not smaller than ε contains an x with ‖x‖ ≤ 1− δ. It is well known (cf. [2]) that X is NUC iff X is reflexive and UKK, where X is said to be uniformly Kadec-Klee, abbreviated UKK, if: for every ε > 0 there exists δ > 0 such that ‖x‖ ≤ 1− δ whenever x is the weak limit of a sequence in BX with separation not smaller than ε. This characterization motivates the following definition of the function ∆X : [0, 2]→ [0, 1] that we call the KK-modulus of X: ∆X(ε) = inf{1− ‖x‖ | xn w −→ x, {xn} ⊂ BX , sep{xn} ≥ ε}. Here, we agree to define ∆X(ε) = 1 if the right hand side infimum ranges over the empty set (i.e., we take the infimum in the ordered set [0, 1]). This means, that in case of a finite dimensional space X, we have