Monotonicity Properties of Musielak–Orlicz Spaces and Dominated Best Approximation in Banach Lattices
✍ Scribed by H Hudzik; W Kurc
- Publisher
- Elsevier Science
- Year
- 1998
- Tongue
- English
- Weight
- 288 KB
- Volume
- 95
- Category
- Article
- ISSN
- 0021-9045
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✦ Synopsis
Criteria for strict monotonicity, lower local uniform monotonicity, upper local uniform monotonicity and uniform monotonicity of a Musielak Orlicz space endowed with the Amemiya norm and its subspace of order continuous elements are given in the cases of nonatomic and the counting measure space. To complete the results of Kurc (J. Approx. Theory 69 (1992), 173 187), criteria for upper local uniform monotonicity of these spaces equipped with the Luxemburg norm are also given. Some applications to dominated best approximation are presented.
1998 Academic Press
1. PRELIMINARIES
In the following, X always denotes a Banach lattice with a lattice norm &} &. Following [6] recall that X is said to be uniformly monotone (UM) (UMB in [5]) if for every =>0 there exists ;(=)>0 such that & f+ g&>1+;(=) whenever f, g # X + (the positive cone in X), & f &=1 and &g& =. It is known (see [23]) that X is UM if and only if for every =>0 there exists '(=)>0 such that & f& g& 1&'(=) whenever f g, f, g # X + , & f &=1 and &g& =. X is said to be strictly monotone (STM) if & f& g&<& f & whenever f g 0 and g{0. From the characterizations of local uniform monotonicity for E 8 and h 8 given in this paper it follows that this property must in general be split into the ULUM and LLUM (LUM in [23]) properties which are defined below. X is said to be upper locally uniformly monotone (ULUM) if for any f # X + with & f &=1