Geometric properties of F-normed Orlicz spaces

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 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.

Some monotonicity properties in F-normed Musielak–Orlicz spaces

“…The results presented in the paper generalize some of the results from papers [3,5], obtained there for Orlicz spaces and their subspaces of order continuous elements endowed with the Mazur-Orlicz F-norm. However, the proofs of the results presented in this paper require deeper techniques then the ones which concern only Orlicz spaces.…”

“…However, the proofs of the results presented in this paper require deeper techniques then the ones which concern only Orlicz spaces. The application of some methods or ideas of proofs from papers [2][3][4][5][6]8] was useful.…”

“…It seems to be interesting to note that the case of Orlicz spaces equipped with the Mazur-Orlicz F-norm already shows that, for instance, the characterization of strict monotonicity and orthogonal strict monotonicity differ (see [3,5]), which was not possible in the norm case.…”

Problems of existence of order copies of $$\ell ^\infty $$ ℓ ∞ and $$L_p(\nu )$$ L p ( ν ) in some non-Banach Köthe spaces

“…Then, for all cases with respect to the kinds of the generating Orlicz functions as well as for both kinds of the measure spaces, the function f Φ,x (•) is continuous on R + if and only if b(Φ) = ∞ and Φ satisfies suitable Δ 2 condition.Proof We can restrict ourselves to x ∈ L Φ (μ)\{0} only. The sufficiency follows from Lemma 4.1 in[9].Necessity. Assuming that either b(Φ) < ∞ or b(Φ) = ∞ and Φ does not satisfy the suitable condition Δ 2 , one can find x ∈ L Φ (μ)\{0} such that I Φ (x) < ∞ and I Φ (λx) = ∞ for any λ > 1 (see the proofs of Theorems 2 and 3).…”

“…where the operator |T |: L p (ν) → Y fulfills the second identity in (9), and |T | is a lattice isometry), and (iii) The spaces V p := T (L p (ν)) and U p are isometric via a mapping S: V p → U p of the form ST x = |T |x, where S is an isometric involution defined in the proof of Proposition 1.…”