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.
