On some geometric and topological properties of generalized Orlicz–Lorentz sequence spaces

Abstract

Generalized Orlicz-Lorentz sequence spaces λϕ generated by Musielak-Orlicz functions ϕ satisfying some growth and regularity conditions (see [28] and [33]) are investigated. A regularity condition δ λ 2 for ϕ is defined in such a way that it guarantees many positive topological and geometric properties of λϕ. The problems of the Fatou property, the order continuity and the Kadec-Klee property with respect to the uniform convergence of the space λϕ are considered. Moreover, some embeddings between λϕ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of λϕ, their subspaces of order continuous elements and finite dimensional subspaces are presented. This paper generalizes the results from [19], [4] and [17]. PreliminariesThroughout this paper R, R + and N denote the sets of reals, nonnegative reals and natural numbers, respectively. The triple (N, 2 N , m) stands for the counting measure space, while l 0 = l 0 (m) denotes the space of all sequences x : N → (−∞, ∞). For every x = (x n ) ∞ n=1 ∈ l 0 (if it is more convenient the notation x(n) is used in place of x n ) we define supp x = {n ∈ N : x n = 0} and |x|(n) = |x(n)| for all n ∈ N , that is, |x| denotes the absolute value of x.The letter e stands for a Banach sequence lattice over the measure space (N, 2 N , m), that is, e is a nontrivial Banach subspace of l 0 (that is e = {0}) satisfying the following conditions (see [23] and [27]):(i) if x ∈ e, y ∈ l 0 and |y| ≤ |x| (i.e., |y n | ≤ |x n | for all n ∈ N ), then y ∈ e and y ≤ x .(ii) there exists a sequence x in e that is strictly positive on the whole N . The positive cone in e is denoted by e + . A Banach lattice e is said to have the Fatou property (e ∈ (FP) for short) if for any x ∈ l 0 and (x m ) ∞ m=1 in e + such that x m x coordinatewise and sup m x m < ∞, we have x ∈ e and x = lim m x m (see [23] and [27]).An element x ∈ e is said to be order continuous if for any sequence (x m ) in e + such that x m ≤ |x| and x m → 0 coordinatewise there holds x m → 0. The subspace e a of all order continuous elements in e is an order ideal of e. A Banach sequence lattice e is said to be order continuous (e ∈ (OC) for short) if e a = e (see [23] and [27]). Notice that e a satisfies condition (ii) from the definition of a Banach sequence lattice, that is, supp e a = N .We say that e has the coordinatewise Kadec-Klee property (e ∈ (H c ) for short) if for any x ∈ e and any sequence (x m ) in e the conditions x m → x and x m (n) → x(n) for all n ∈ N , yield x − x m → 0. We *

Kadec-Klee properties of Calderón-Lozanovskiĭ sequence spaces

“…Proof The sufficiency has been proved in [7,Theorem 4.4] but under general assumption that ϕ < ∞. However, recall that, if E → l ∞ and ϕ ∈ 2 (0) then, for any x ∈ E ϕ , the equivalence I ϕ (x) = 1 ⇔ x ϕ = 1 holds if and only if ϕ(b ϕ ) inf i e i e ≥ 1 (Lemma 1.4(ii) from [12]).…”

“…This property, also called the Radon-Riesz property or property H , has been considered in many classes of Banach spaces (see [1,2,5,7,15]). If we consider E more generally over σ −finite and complete measure space (T, , μ) and we replace the weak convergence σ (E, E * ) by the convergence in measure (x n μ → x), by the convergence in measure on every set of finite measure (x n μ → x locally) or by the uniform convergence (x n ⇒ x), then we say that E has the Kadec-Klee property with respect to convergence in measure, local convergence in measure or uniform convergence, respectively (we shall write…”

“…Thus E → c 0 { e n }. Thus, by (+), we may apply Lemma 2.9 from [12] and Theorem 2.4 from [7] to obtain that ϕ ∈ E 2 . The sufficiency is done by Corollary 12 from [14].…”

“…The Calderón-Lozanovskiȋ spaces are one of important classes of Banach lattices, especially due to their application in the interpolation theory. Geometry of Calderón-Lozanovskiȋ spaces has been deeply studied recently (see for example [3,[7][8][9][10]12,14]). …”

“…Here we consider the respective sequence case. Some partial results concerning Kadec-Klee property with respect to pointwise convergence in generalized Calderón-Lozanovskiȋ and Orlicz-Lorentz sequence spaces have been presented in [7] and [2]. We present full characterizations of Kadec-Klee properties with respect to pointwise convergence (with respect to uniform convergence) in Calderón-Lozanovskiȋ sequence spaces.…”