Geometry of norm attainability in Orlicz spaces

Evans, Mogoi N.; Obogi, Robert

doi:10.30495/maca.2025.2052193.1127

Geometry of norm attainability in Orlicz spaces

10.30495/maca.2025.2052193.1127

Volume 7, Issue 1
Winter 2025

Pages 71-78

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

This paper investigates norm attainability and modular properties in Orlicz spaces, which generalize $L^p$-spaces and are key in functional analysis and nonlinear problems. It presents theorems on norm attainment, orthogonality, weak compactness, and uniform convexity, and introduces a novel criterion connecting the convexity of the Orlicz function with the smoothness and reflexivity of the space. The research extends classical concepts such as the $\Delta_2$-condition to ensure completeness and separability. The results have practical applications in nonlinear optimization, variational analysis, machine learning, signal processing, image reconstruction, and solving PDEs with nonlinear boundary conditions, providing a strong foundation for future research in these areas.

Keywords

Orlicz Spaces

Norm Attainability

Modular Properties

Convexity

Nonlinear Optimization

Duality Theory

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