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

Jiaqi Dong,Yunan Cui,Marek Wisła,Seppo Hassi

doi:10.1155/2022/5902635

Abstract

Using some new techniques, the necessary and sufficient conditions for Kadec-Klee property of Orlicz function spaces equipped with s-norms are presented. An original method that was used in the process of inquiry and the obtained results also systematically complete and broaden the characterization of Kadec-Klee property of Orlicz spaces.

Highlights

Introduction and PreliminariesOrlicz spaces, introduced by W
On Orlicz spaces of measurable functions, the classical Orlicz and Luxemburg norm can be defined by use of the Amemiya formula: ‖x‖oΦ infk>0(1/k)
Wisła presented a universal and general method of introducing norms (s-norms) in Orlicz spaces in 2019, and the introduction of Orlicz spaces equipped with s-norms covers all the cases mentioned above

Summary

Introduction and Preliminaries

Orlicz in 1932, form a wide class of Banach spaces of measurable functions (in the case of atomless measure) or sequences (in the case of counting measure) (see [1]). On Orlicz spaces of measurable functions, the classical Orlicz and Luxemburg norm can be defined by use of the Amemiya formula: ‖x‖oΦ infk>0(1/k). Σ-measurable real functions defined on G. for all u, v ∈ R. Let s be an outer function and Φ be an Orlicz function. S∗ is an outer function that is conjugate to s in the Holder sense. For an outer function s and its righthand derivative s+′, define v ω(v) 􏽚 s′−+1(t) dt,. We say that an Orlicz function Φ satisfies condition Δ2 (Φ ∈ Δ2, for short) if there exists K > 0 and u0 > 0, such that.

Main Results

Using quality

Ga k dt