Banach Spaces and Complete Inner Product Spaces

Abstract

The fundamental work of the Italian mathematician Vito Volterra (from 1887 onwards when he published his earliest paper ‘Functions Depending on Other Functions’) has shown a definite advantage of the operational techniques in the study of families of functions. Later work in this direction by Ivar Fredholm, Frigyes Riesz, E. Hellinger and O. Toeplitz (as well as many others) emphasized the direction inaugurated by Volterra. This work led to the consideration of some classes of spaces which are of fundamental importance. First was considered by Stefan Banach, E. Helly, H. Hahn and Norbert Wiener (independently of each other) the class of normed linear spaces (1922). It is worth mentioning that the axioms for linear spaces as well as some notions about operators on such spaces, are given in the book by Salvatore Pincherle (1901; in collaboration with Ugo Amaldi) in a form almost identical with that now in use.*