A comparative study of soft 2-norms: Limitations, refined concepts, and structural integration

Abstract

The present work aims to further contribute to the study of soft linear structures by analyzing key definitions and providing illustrative examples that highlight both the strengths and limitations of existing approaches. In particular, similarly to classical linear functional analysis, we establish a connection between the refined soft 2-norm and the soft norm by means of a suitable soft basis. This relationship clarifies the structural coherence between these two notions and supports the development of a consistent mathematical framework within soft set theory. Refined soft 2-norm studies contribute to a globally consistent way to measure linear dependence across all parameters, establish connections with soft 2-inner products, and provide concrete constructions that extend classical 2-norm theory to soft vector spaces with parametric structure. In this work, a careful comparison between the soft 2-norm and the refined soft 2-norm is presented to highlight their conceptual and structural differences. Furthermore, while both norms were attempted to be related to Gram-type 2-inner products, the refined soft 2-norm was found to naturally provide a consistent derivation from the soft 2-inner product thanks to its parameter-coordinated structure. Therefore, the refined version not only strengthens the link between geometry and algebra in soft vector spaces, but also fills a gap in the literature by providing an example where global parametric dependence is clearly characterized. Another aim of this paper is to investigate an extension theorem for soft linear functionals in refined soft 2-normed spaces. By introducing an induced soft norm obtained from a fixed soft vector, we establish a Hahn--Banach type extension theorem in the framework of refined soft 2-normed spaces. Furthermore, we discuss consequences related to soft bounded linear functionals and the dual structure of these spaces. Our results reinforce the importance of refined concepts in soft set theory and contribute to the ongoing effort to establish a solid and coherent mathematical framework for handling uncertainty.