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

Abstract

We investigate norm attainability and duality properties in Orlicz spaces, extending classical results from Banach and Hilbert spaces to a more gen eral functional framework. We establish 14 fundamental theorems that character ize norm attainment in terms of strict convexity, uniform convexity, and weak con vergence. We explore the duality structure of Orlicz spaces, highlighting key differ ences from Lp 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 con tribute to a deeper understanding of Orlicz space geometry and its implications for functional analysis and applied mathematics.