Some fundamental geometric and topological properties of generalized Orlicz-Lorentz function spaces

✍ Scribed by Paweł Foralewski

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 231 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200810083

✦ Synopsis

Abstract

Generalized Orlicz‐Lorentz function spaces Λ~φ~ generated by Musielak‐Orlicz functions φ satisfying some growth and regularity conditions (cf. 34 and 38) are investigated. A regularity condition \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Delta ^{\Lambda }_{2}$\end{document} for φ is defined in such a way that it guarantees many positive topological and geometric properties of Λ~φ~. The problems of the Fatou property, order continuity (separability) and the Kadec‐Klee property with respect to the local convergence in measure of Λ~φ~ are considered. Moreover, some embeddings between Λ~φ~ and their two subspaces are established and strict monotonicity as well as lower and upper local uniform monotonicities are characterized. Finally, necessary and sufficient conditions for rotundity of Λ~φ~ are presented. This paper generalizes the results from 20. Analogous results in the sequence case were presented in 10 and 11, but the techniques in the function case are different. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim