Functional analysis: a historical perspective. (Spanish: Análisis Funcional: una perspectiva histórica)

Abstract

This interesting, well-written, informative article presents the origin of functional analysis and reviews the highlights of its history till 1960. The strategy of the paper is to present the problems which lead to new concepts and motivated higher levels of abstraction. As classical antecedents of functional analysis, the method of separation of variables of Fourier, the problem of Sturm-Liouville, the calculus of variations and the problem of Dirichlet are mentioned. The work of Volterra and Fredholm on integral equations is analysed. The very important contributions of Hilbert, his student Schmidt, E. Fischer and F. Riesz are presented in detail. It is clear that the author has studied the original articles quoted in these sections. The next chapter considers the development of Banach space theory, and explains the work of Fréchet, Mazur, Steinhaus and Banach himself. Von Neumann's application of Hilbert space theory to quantum mechanics, and Gelfand's work on Banach algebras are discussed. The last section reports on the work by Kolmogorov, Köthe, Toeplitz, Mackey, Dieudonné, L. Schwartz and Grothendieck about the theory of locally convex spaces and its applications to the theory of distributions and the kernel theorem.