ABSTRACT

This chapter develops neoclassical analysis in the framework of hyperseminormed and hypernormed vector spaces. Neoclassical analysis extends methods of classical calculus reflecting imprecision and uncertainties that arise in computations and measurements. In it, ordinary structures of analysis, that is, functions, sequences, series, and operators, are studied by means of fuzzy concepts such as fuzzy limits, fuzzy continuity, and fuzzy derivatives. For instance, continuous operators, which are studied in the classical functional analysis, become a part of the set of the fuzzy continuous operators studied in neoclassical analysis. Aiming at representation of uncertainty, vagueness and imprecision, neoclassical analysis makes, at the same time, methods of the classical calculus more precise with respect to real life applications and extends the scope of the classical calculus and functional analysis. Fuzzy continuity extends the concept of continuity for functions, functionals and operators. The chapter introduces and explores twofold fuzzy continuity for functionals and operators in the new realm of hyperseminormed vector spaces and algebras.