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.
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Dedicated to Professor Karol Baron on the occasion of his 70th Birthday.
Yunan Cui gratefully acknowledges the support of NFS of CHINA (11871181).
Paweł Kolwicz was supported by the Ministry of Science and Higher Education of Poland, Grant Number 04/43/DSPB/0094.
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Cui, Y., Hudzik, H., Kaczmarek, R. et al. Geometric properties of F-normed Orlicz spaces. Aequat. Math. 93, 311–343 (2019). https://doi.org/10.1007/s00010-018-0615-y
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DOI: https://doi.org/10.1007/s00010-018-0615-y
Mathematics Subject Classification
Keywords
- Orlicz spaces
- Mazur–Orlicz F-norm
- Köthe normed spaces
- F-normed Köthe spaces
- Symmetric spaces
- Symmetric F-normed spaces
- Order continuity
- Fatou properties
- Strict monotonicity
- Orthogonal strict monotonicity
- Lower local uniform monotonicity
- Orthogonal lower local uniform monotonicity
- Upper local uniform monotonicity
- Orthogonal upper local uniform monotonicity
- Uniform monotonicity
- Orthogonal uniform monotonicity
- Condition \(\Delta _2\)
- Strong condition \(\Delta _2\)
- Kadec–Klee properties \(H_{l}, H_{g}, H_{u}\) and \(H_{c}\)