On Parseval’s Identity and Plancherel Relation in 2-Inner Product Spaces

Abstract

This paper investigates the foundational concepts of linear 2-normed spaces and 2-inner product spaces, presenting essential definitions and theorems that elucidate their structure. We establish Parseval’s identity for finite orthonormal sets within these spaces and examine its relationship with the Plancherel relation. Our findings include theorems related to orthonormal sets and Bessel's inequality, highlighting their implications for countably infinite orthonormal sets. By contributing to the theoretical framework of 2-inner product spaces, we enhance the understanding of orthogonality and summability in this context. Introduced by Diminnie et al. (1973), the 2-inner product extends traditional inner product notions into a two-dimensional framework, offering valuable insights into unique mathematical structures. Recent contributions, such as the formalization of orthonormal sets by Elumalai and Patricia (2000) and the rigorous proofs of the Riesz theorems by Harikrishnan et al. (2011), further deepen our understanding of these spaces. This paper aims to develop important inequalities based on orthonormal sets and present equivalent statements related to them, fostering continued research in various mathematical contexts.