Lower and upper local uniform $K$ -monotonicity in symmetric spaces

Abstract

Using the local approach to the global structure of a symmetric space E we establish a relationship between strict K-monotonicity, lower (resp. upper) local uniform K-monotonicity, order continuity and the Kadec-Klee property for global convergence in measure. We also answer the question under which condition upper local uniform K-monotonicity concludes upper local uniform monotonicity. Finally, we present a correlation between K-order continuity and lower local uniform K-monotonicity in a symmetric space E under some additional assumptions on E.2010 Mathematics Subjects Classification: 46E30, 46B20, 46B42.

Relationships between K-monotonicity and rotundity properties with application

“…Now, since E d is compactly fully k-rotund and strictly Kmonotone, by Theorem 4.9 we have E is upper locally uniformly K-monotone. Hence, since E is order continuous, by ( 20) and ( 23) as well as by Theorem 3.13 in [5] we conclude y * n − x * n E → 0, which completes the proof.…”

“…Immediately, using the same technique as in the proof of Theorem 2.10 in [4] and in view of Corollary 1.6 and Proposition 1.7 in [4] we get (i) ⇔ (ii). In consequence, by Theorem 3.13 in [5] we have (ii) ⇔ (iii) ⇔ (iv). Now, according to Theorem 3 in [18] and by Theorem 6.1 we present a correspondence between approximative compactness and K-monotonicity properties in symmetric spaces.…”

“…for any k ∈ N. Then, for any subsequence (x n,1 ) of (x n ), by (4) we have (5) x n,1 + x X → 2 and x n,1 X → 1.…”

“…Next, passing to subsequence (u n,1 ), by the triangle inequality of the norm in X and by (5) and (6) it follows that u n,1 + x X → 2.…”

“…Recently, in the papers [13,18], there have been shown, among others, a complete correspondence between fully k-rotundity properties, rotundity and reflexivity with an application to the approximation theory in Banach spaces. The next interesting results were published in [3], where authors have investigated, inter alia, rotundity properties on E d the positive cone of all nonnegative and decreasing elements of a K-monotone symmetric space E. The further motivation of our investigation can be found in [5,6,7,11], where authors have presented a correspondence and complete criteria for K-order continuity, strict K-monotonicity and uniform K-monotonicity properties with application to the best dominated approximation problems in the sense of the Hardy-Littlewood-Pólya relation. The main idea of this paper is to find a relationship between reflexivity, fully k-rotundity properties and uniform K-monotonicity properties with application to the approximation theory.…”