Kadec–Klee property with respect to the local convergence in measure of Orlicz–Lorentz spaces
Corresponding Author
Paweł Foralewski
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego, Poznań, Poland
Correspondence
Paweł Foralewski, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland.
Email: [email protected]
Search for more papers by this authorJoanna Kończak
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego, Poznań, Poland
Search for more papers by this authorCorresponding Author
Paweł Foralewski
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego, Poznań, Poland
Correspondence
Paweł Foralewski, Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego 4, 61-614 Poznań, Poland.
Email: [email protected]
Search for more papers by this authorJoanna Kończak
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Uniwersytetu Poznańskiego, Poznań, Poland
Search for more papers by this authorAbstract
In this paper, we find criteria for the Kadec–Klee property with respect to the local convergence in measure in both Orlicz–Lorentz spaces as well as their subspaces of order continuous elements. In the case of Orlicz norm, the presented results are new, whereas in the case of Luxemburg norm, we rely heavily on known results, which we show for the first time as a whole. Finally, we apply the obtained results to Orlicz spaces.
REFERENCES
- 1C. Bennett and R. Sharpley, Interpolation of operators, Pure Appl. Math. 129 (1988), 1–469.
- 2S. T. Chen, Geometry of Orlicz spaces, Dissertationes Math. (Rozprawy Mat.) 356 (1996), 204.
- 3V. I. Chilin, P. G. Dodds, A. A. Sedaev, and F. A. Sukochev, Characterisations of Kadec-Klee properties in symmetric spaces of measurable functions, Trans. Amer. Math. Soc. 348 (1996), 4895–4918.
- 4Y. Cui, P. Foralewski, and H. Hudzik, -constants in Orlicz-Lorentz function spaces, Math. Nachr. 292 (2019), 2556–2573.
- 5Y. Cui, P. Foralewski, and J. Kończak, Orlicz-Lorentz sequence spaces equipped with the Orlicz norm, Acta Math. Sci. Ser. B (Engl. Ed.) 42 (2022), 623–652.
- 6T. Dominguez, H. Hudzik, G. López, M. Mastyło, and B. Sims, Complete characterizations of Kadec-Klee properties in Orlicz spaces, Houston J. Math. 29 (2003), 1027–1044.
- 7P. Foralewski, Some fundamental geometric and topological properties of generalized Orlicz-Lorentz function spaces, Math. Nachr. 284 (2011), 1003–1023.
- 8P. Foralewski, H. Hudzik, and P. Kolwicz, Non-squareness properties of Orlicz-Lorentz sequence spaces, J. Funct. Anal. 264 (2013), 605–629.
- 9P. Foralewski and J. Kończak, Orlicz-Lorentz function spaces equipped with the Orlicz norm, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 117 (2023), Paper No. 120, 23.
- 10W. Z. Gong and D. X. Zhang, Monotonicity in Orlicz-Lorentz sequence spaces equipped with the Orlicz norm, Acta Math. Sci. Ser. B (Engl. Ed.) 36 (2016), 1577–1589.
- 11H. Hudzik and M. Mastyło, Strongly extreme points in Köthe-Bochner spaces, Rocky Mountain J. Math. 23 (1993), 899–909.
- 12A. Kamińska, Some remarks on Orlicz-Lorentz spaces, Math. Nachr. 147 (1990), 29–38.
- 13A. Kamińska, K. Leśnik, and Y. Raynaud, Dual spaces to Orlicz-Lorentz spaces, Studia Math. 222 (2014), 229–261.
- 14L. V. Kantorovich and G. P. Akilov, Functional analysis, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from Russian by Howard L. Silcock.
- 15M. A. Krasnoselskiĭ and Ja. B. Rutickiĭ, Convex functions and Orlicz spaces, P. Noordhoff Ltd., Groningen, 1961. Translated from the first Russian edition by Leo F. Boron.
- 16S. G. Kreĭn, Yu. Ī. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs, vol. 54, American Mathematical Society, Providence, R.I., 1982.
- 17F. E. Levis and H. H. Cuenya, Gateaux differentiability in Orlicz-Lorentz spaces and applications, Math. Nachr. 280 (2007), 1282–1296.
- 18J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. II, Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97, Springer-Verlag, Berlin-New York, 1979.
10.1007/978-3-662-35347-9 Google Scholar
- 19L. Maligranda, Indices and interpolation, Dissertationes Math. (Rozprawy Mat.) 234 (1985), 49.
- 20L. Maligranda, Orlicz spaces and interpolation, Seminários de Matemática [Seminars in Mathematics], vol. 5, Universidade Estadual de Campinas, Departamento de Matemática, Campinas, 1989.
- 21S. J. Montgomery-Smith, Comparison of Orlicz-Lorentz spaces, Studia Math. 103 (1992), 161–189.
- 22J. Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
- 23J. Wang and Y. Chen, Rotundity and uniform rotundity of Orlicz-Lorentz spaces with the Orlicz norm, Houston J. Math. 38 (2012), 131–151.
- 24J. Wang and Z. Ning, Rotundity and uniform rotundity of Orlicz-Lorentz sequence spaces equipped with the Orlicz norm, Math. Nachr. 284 (2011), 2297–2311.
