ASSESSING THE FUZZY N-INNER PRODUCT SPACES AND ITS IMPACT ON LINEAR OPERATORS

Resumo

Fuzzy n-inner product space “( )” and its concept induced fuzzy n-normed linear space “( )”, which generalizes the classical n-inner product spaces to the fuzzy setting, and investigates what effects these generalized fuzzy structures have on the properties of linear operators. We construct a specific  and obtain some fundamental properties that relate the fuzzy inner product to the fuzzy norm. Then, the paper characterizes fuzzy bounded and fuzzy continuous linear operators  between these fuzzy spaces. One of the surprising findings is that, in this fuzzy context, a fuzzy bounded linear operator is always fuzzy continuous; however, the converse may not be true in general, which represents a critical difference from the classic theory of operators. Nevertheless, we prove that in the case when the space  is finite-dimensional, fuzzy continuity yields fuzzy boundedness. Finally, this study generalizes basic concepts from classical operator theory, such as adjoint, self-adjoint, normal, and unitary operators, to this new fuzzy context. With this work, a solid framework for the study of linear operators in imprecise contexts is laid, which provides the possibility for investigation into areas such as fuzzy differential equations and signal processing under uncertainty. Through a rigorous investigation of operator behavior in these uncertain situations, it represents a significant contribution to the growing field of fuzzy mathematics. We provide a basis for possible applications in fuzzy modeling, uncertain signal processing, and multidimensional decision-making systems.