Absolute sums of Banach spaces and some geometric properties related to rotundity and smoothness

Abstract

We study the notions of acs, luacs and uacs Banach spaces which were introduced in [26] and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested in (mainly infinite) absolute sums of such spaces. We also introduce some new classes of spaces that lie inbetween those of acs and uacs spaces and study their behaviour under the formation of absolute sums as well.x n + x → 2 ⇒ x n → x weakly,

Köthe-Bochner Spaces and Some Geometric Properties Related to Rotundity and Smoothness

“…Now let us treat the case of uacs spaces. In analogy to [11,Definition 3.15] we say that an order continuous Köthe function space E has property (u + ) if for every ε > 0 there is some δ > 0 such that for all f, g ∈ S E and every h ∈ S E ′ we have…”

“…Note that since δ X uacs is continuous on (0, 1) (see [6,Lemma 3.10] or [11,Lemma 2.11]), the function α is measurable. Using (3.86) it is easy to see that f (t) + g(t) ≤ 2(1 − α(t))β(t) a. e. (3.87) By (3.84) and (3.85) we have S γ(t) dµ(t) ≤ 2.…”

“…Now let us assume that X is msluacs and E * has the Kadets-Klee* property and weak*-sequentially compact unit ball. To show that E(X) is msluacs we will proceed in an analogous way to the proof of [11,Proposition 4.7], which in turn uses techniques from the proof of [8,Proposition 4]. So let us take two sequences (f n ) n∈N , (g n ) n∈N in S E(X) and f ∈ S E(X) such that f n + g n − 2f E(X) → 0.…”

“…Since X is wuacs the dual space X * is sluacs (cf. [11,Proposition 2.16]) and thus (because of x * n (x n ) → 1) we can conclude y * (x n ) → 1, whence y * (y) = 1 = y * (x), which implies x + y = 2, which by the rotundity of X implies x = y. (i) If X is acs then X is luacs.…”

WORTH property, García-Falset coefficient and Opial property of infinite sums

“…By Corollary 3.3 we have R(X 1 ) < 2 and by Example 3.4 we have R(X 3 ) < 2. Also, by [11,Corollary 3.17] and the remarks after [11, Definition 1.5] X 2 is again a U -space, so R(X 2 ) < 2. From the aforementioned result [4,Theorem 7] it follows that R(X 4 ) < 2 and since Theorem 7] implies that R(X) < 2.…”