Über die weitere Entwicklung der Funktionalanalysis bis 1932. (Further development of functional analysis up to 1932) (Q1078470)

In continuation of an earlier paper [ibid. 41, 25-35 (1986; review above)] this paper investigates the development of basic ideas in functional analysis from 1906 to 1932. It is shown that Fréchet's dissertation of 1906 initiates the development of ideas related to metric space and topological space (Riesz 1906 and 1908, Hausdorff 1914, etc.). 1906 is also the year of the earliest functional analytic treatment of integral equations (Hilbert), leading to Hilbert space, the Riesz-Fischer theorem (1907), spectral theory of operators on Hilbert space and further developments on functionals, the evolution of the theory of normed spaces (1916-1932, F. Riesz, Banach, Wiener, Steinhaus, etc.). The paper places much emphasis on F. Riesz's work and on the effect of quantum mechanics which, through the work of J. v. Neumann and M. H. Stone, led to a unification and establishment of functional analysis as a field of its own. The paper does not consider developments which began before 1932 but culminated only later, for reasons of the length of the article (fixed point theorems, ergodic theory, abstract probability etc.).