Geometric properties of F-normed Orlicz spaces

Henryk Hudzik,Paweł Kolwicz,Yunan Cui,Radosław Kaczmarek

doi:10.1007/s00010-018-0615-y

Abstract

The paper deals with F-normed functions and sequence spaces. First, some general results on such spaces are presented. But most of the results in this paper concern various monotonicity properties and various Kadec–Klee properties of F-normed Orlicz functions and sequence spaces and their subspaces of elements with order continuous norm, when they are generated by monotone Orlicz functions on {mathbb {R}}_{+} and equipped with the classical Mazur–Orlicz F-norm. Strict monotonicity, lower (and upper) local uniform monotonicity and uniform monotonicity in the classical sense as well as their orthogonal counterparts are considered. It follows from the criteria that are presented for these properties that all the above classical monotonicity properties except for uniform monotonicity differ from their orthogonal counterparts [in contrast to Köthe spaces (see Hudzik et al. in Rocky Mt J Math 30(3):933–950, 2000)]. The Kadec–Klee properties that are considered in this paper correspond to various kinds of convergence: convergence locally in measure and convergence globally in measure for function spaces, uniform convergence and coordinatewise convergence in the case of sequence spaces.

Highlights

The geometry of Banach spaces has found a lot of applications and has been intensively developed during the last decades
It is known that monotonicity properties play an analogous role in the best dominated approximation problems in Banach lattices as the respective rotundity properties do in the best approximation problems in Banach spaces
: (i) The Orlicz space LΦ(μ) is strictly monotone if and only if b(Φ) = ∞, Φ is strictly increasing on R+ and Φ satisfies the condition Δ2(R+) if μ is infinite and the condition Δ2(∞) if μ is finite

Summary

Introduction

The geometry of Banach spaces has found a lot of applications (approximation theory, fixed point theory, ergodic theory, control theory, probability theory, theory of vector analytic functions, theory of generalized inverses) and has been intensively developed during the last decades. It is known that monotonicity properties (strict and uniform monotonicity) play an analogous role in the best dominated approximation problems in Banach lattices as the respective rotundity properties (strict and uniform rotundity) do in the best approximation problems in Banach spaces (see [7,33]). The various monotonicity properties and their applications to dominated best approximation and complex rotundities have been intensively investigated by many mathematicians (see [6,7,9,16,20,22,29,30,33,34]). It is worth mentioning that Kadec–Klee properties are applicable in the best dominated approximation problems in Banach lattices (see [8] and the references therein). Our paper offers an extension of results on monotonicity and Kadec–Klee properties of Banach function lattices to F -normed function lattices, to F -normed Orlicz spaces (compare with [5,8,9,10,11,12,19,21,22,24,26,27,28,33, 34])

Preliminaries

F -normed Kothe spaces

F -normed Orlicz spaces

Results on general F-normed Kothe spaces

Results on F-normed Orlicz-spaces