Mathematical Analysis and its Contemporary Applications

Norm attainment and structural properties in Orlicz spaces: A comprehensive study on strict convexity, duality, and optimization

10.30495/maca.2025.2056607.1133
Volume 7, Issue 3
Summer 2025
Pages 1-12
Mathematical Analysis and its Contemporary Applications

Document Type : Original Article

Authors

Mogoi N. Evans 1
Robert Obogi 2

1 Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Kenya

2 Department of Mathematics and Actuarial Science, Kisii University, Kenya

Abstract
We investigate norm attainability and duality properties in Orlicz spaces, extending classical results from Banach and Hilbert spaces to a more general functional framework. We establish 14 fundamental theorems that characterize norm attainment in terms of strict convexity, uniform convexity, and weak convergence. We explore the duality structure of Orlicz spaces, highlighting key differences from $L^p$ spaces and providing a variational characterization of the norm. We also discuss applications in optimization and variational problems, demonstrating the significance of norm-attaining functionals in these settings. Our findings contribute to a deeper understanding of Orlicz space geometry and its implications for functional analysis and applied mathematics.

Keywords

Orlicz spaces
norm attainability
duality properties
convexity
functional analysis
[1] L. Bernal-Gonzalez, D. L. Rodriguez-Vidanes, J. B. Seoane-Sepulveda, and H.-J. Tag, New results in analysis of Orlicz-Lorentz spaces, arXiv preprint arXiv:2312.13903 (2023).
[2] N. Dunford, Uniformity in linear spaces, Trans. Amer. Math. Soc. 44(2) (1938), 305–356. 
[3] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, 1965.
[4] A. Kaminska and H.-J. Tag, On Kothe duals of Orlicz-Lorentz spaces, arXiv preprint arXiv:2404.07368 (2024).
[5] M. A. Krasnosel’skii and Ya. B. Rutickii, Convex Functions and Orlicz Spaces, Gordon & Breach, 1961.
[6] M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, 1991.
[7] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20(11) (1971), 1077–1092.
[8] J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, 1983.
[9] B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc., 44(2) (1938), 277–304.
[10] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.
[11] A. P. Robertson, On unconditional convergence in topological vector spaces, Proc. Royal Soc. Edin. A, 69 (1969), 113-125.
[12] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17(4) (1967), 473–483.
[13] A. Zygmund, Trigonometric Series, Volume 1, Cambridge University Press, 1959.
  • Receive Date 25 March 2025
  • Accept Date 09 June 2025

Home

Submit Manuscript

Reviewers

Contact Us