On geometric structure of symmetric spaces

Abstract

In this article we discuss local approach to strict K-monotonicity and local uniform rotundity in symmetric spaces. We prove several general results on local structure of symmetric spaces E showing relation between strict monotonicity and strict K-monotonicity and the Kadec-Klee property for global convergence in measure. We also present the full criteria for points of upper K-monotonicity in Lorentz spaces Γ p,w for degenerated weight function w. Next we characterize local uniform rotundity in symmetric spaces E proving several correspondences between x ∈ E a point of local uniform rotundity and its decreasing rearrangement x * and absolute value |x|. Finally, we apply these results to find complete criteria for local uniform rotundity of Lorentz spaces Γ p,w .

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

“…Immediately, in view of Remark 3.1 in [8], by Proposition 3.4 and Theorems 3.6 and 3.7 we obtain the following results. 1 [0, ∞).…”

“…Let x, x n ∈ E for any n ∈ N, x * * n → x * * in measure and x n E → x E . Now, proceeding analogously as in the proof of Theorem 3.8 [8], under the assumption that x is an H g point and x * (∞) = 0, in view of Theorem 3.3 [11] we complete the proof. (ii) ⇒ (iii).…”

“…It is also well known that by the Lions-Peetre K-method (see [2,17]), the space Γ p,w is an interpolation space between L 1 and L ∞ . For more details about the properties of the spaces Λ p,w and Γ p,w the reader is referred to [8,10,11,15,16].…”

“…Authors presented a complete characterization of U LU KM written in terms of strict Kmonotonicity and the Kadec-Klee property for global convergence in measure in symmetric spaces, among others. Recently, many interesting results have appeared in [4,8,11,12,14], where there have been explored the global and local K-monotonicity structure of Banach spaces.…”

Strict K-monotonicity and K-order continuity in symmetric spaces

“…The next motivating research was published in [9], where there has been established among others a connection between the best dominated approximation and the KadecKlee property for global convergence in measure in Banach function spaces. Recently, in view of the previous investigation there appeared many results [5][6][7]10,14] devoted to exploration of the global and local monotonicity and rotundity structure of Banach spaces applicable in the best approximation problems. The main inspiration for this article appeared in paper [4], where there has been introduced a new type of the best dominated approximation with respect to the Hardy-Littlewood-Pólya relation ≺.…”