Kadec–Klee properties of Orlicz–Lorentz sequence spaces equipped with the Orlicz norm

Abstract

The necessary and sufficient conditions for both the Kadec–Klee property as well as the Kadec–Klee property with respect to the coordinatewise convergence in Orlicz–Lorentz sequence spaces equipped with the Orlicz norm and generated by arbitrary Orlicz functions as well as any non-increasing weight sequences are given. Moreover, for their subspaces of elements with an order continuous norm the full characterization of the Kadec–Klee property with respect to the coordinatewise convergence is presented. Some tools useful in the proofs of the main results are also provided.

Extreme points in Orlicz spaces equipped with s‐norms and closedness

“…In this section, we will give the description of extreme points of the unit ball in Orlicz spaces equipped with s ‐norms. Criteria for extreme points in Orlicz spaces equipped with the Orlicz norm [5, 6, 13, 17], the Luxemburg norm [2, 17], and the p ‐Amemiya norm with $$1\leqslant p \leqslant \infty$$ [8] have been found. Our results will unify and extend these three classical cases.…”

“…The theory of Orlicz spaces has been greatly developed because of its important theoretical properties and value in applications. Up to now, criteria that an element in the unit sphere of an Orlicz space equipped with the Orlicz norm, the Luxemburg norm, and the p ‐Amemiya norm is an extreme point have been obtained [5, 6, 8, 13, 17]. In [23], using the concept of an outer function, Wisła presented a general and universal method of introducing norms in Orlicz spaces that covered the classical Orlicz and Luxemburg norms, and p ‐Amemiya norms ($$1 \leqslant p \leqslant \infty$$).…”

Kadec-Klee Property in Orlicz Function Spaces Equipped with S-Norms

“…Let X be a Banach space. If x n ∈ S(X), x ∈ S(X), and x n ⟶ w x imply x n ⟶ x(n ⟶ ∞), then we say that X has the Kadec-Klee property (see [5][6][7]).…”