ABSTRACT
In this chapter, the author advances the theory of extrafunctions developed in Burgin extending it to the theory of hyperoperators and hyperfunctionals, which includes the classical theory of operators and functionals. The usage of the word functional originated in the calculus of variations, where a functional was comprehended as a function with functions as its arguments and numbers as its values. The general concept of a functional was introduced by the Italian mathematician Volterra, who used the term functions of curves or functions of other functions. However, Hadamard was the first to use the name functional. Topological constructions lead to various classes of norm-based real extrafunctions, which include real distributions, extended distributions, hyperdistributions, restricted pointwise extrafunctions, and compactwise extrafunctions. The chapter teaches only extrafunctions that are generated by real functions. Operations of addition and/or subtraction are correctly defined for Q-based hyperoperators.
