Those kinds of infinite direct sums are characterized for which NUC (NUS, respectively) is inheri... more 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 r...
In this paper we extend the result of [6] on the characteristic of convexity of Orlicz spaces to ... more In this paper we extend the result of [6] on the characteristic of convexity of Orlicz spaces to the more general case of Musielak-Orlicz spaces over a non-atomic measure space. Namely, the characteristic of convexity of these spaces is computed whenever the Musielak-Orlicz functions are strictly convex.
Those kinds of infinite direct sums are characterized for which NUC (NUS, respectively) is inheri... more 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 r...
In this paper we extend the result of [6] on the characteristic of convexity of Orlicz spaces to ... more In this paper we extend the result of [6] on the characteristic of convexity of Orlicz spaces to the more general case of Musielak-Orlicz spaces over a non-atomic measure space. Namely, the characteristic of convexity of these spaces is computed whenever the Musielak-Orlicz functions are strictly convex.
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