Analysis of the Hilbertian Properties of the Sobolev Space \(H^1(\Omega)\)

Abstract

This paper thoroughly examines the Hilbertian properties of the Sobolev space H1(\(\Omega\)). Based on the Lebesgue space L2(\(\Omega\)), H1(\(\Omega\)) is of particular importance in the analysis of functions with weak derivatives. The focus is on the Hilbertian structure of this space, which allows for the definition of a specific inner product and a rigorous verification of the associated completeness. These fundamental characteristics facilitate a detailed study of convergence, continuity, and orthogonality of functions in H1(\(\Omega\)), thereby enhancing its relevance for solving various complex mathematical problems. This paper aims to address gaps identified in the existing literature, where the explicit verification of these essential properties is sometimes omitted, thus compromising mathematical rigor and the applicability of results in various contexts.