Properties of Linear Operators on Soft Normed Spaces

Abstract

This paper develops the theory of soft normed spaces and soft linear operators, extending classical functional analysis to the framework of soft sets. We define soft norms and soft operator norms, and explore the properties of soft bounded and soft continuous linear operators. Key results include the equivalence of soft boundedness and soft continuity, and the characterization of the soft operator norm, including positivity, homogeneity, subadditivity, and submultiplicativity. We also show that a linear operator is soft continuous on the entire space if and only if it is soft continuous at a single point. These results provide a rigorous foundation for soft functional analysis, enabling applications in uncertainty modeling, decision-making, and other fields requiring flexible handling of imprecise information.