Book Review: Applied functional analysis (Applications to mathematical physics)

Abstract

Functional analysis is primarily concerned with infinite-dimensional spaces, mainly function spaces-topological vector spaces whose "points" are functions-and mappings between them, usually called operators or, if the range is on the real line or in the complex plane, functionals. Its beginning dates back to 1887, when Volterra published notes in special classes of functionals; but as a large field of its own, it is one of the great mathematical creations of our century. Volterra used concepts from the calculus of variations, which, together with integral equations, supplied motivations, ideas, techniques, and applications during the early period of the development. Its former name, functional calculus (Calcul fonctionnel), coined by Hadamard and Fréchet, indicates that the original purpose was the extension of the calculus to the study of functionals. Cf. [3]. During the nineteenth century, problems and methods on spectral theory of ordinary and partial differential equations, potential theory, Fourier expansions, and special functions had been slowly accumulating. Under the influence of physics the study of "general solutions" of functional equations was gradually superseded by that of solutions satisfying additional conditions.