On the history of ideas of functional analysis (Q1107504)

The paper traces the conceptual development of functional analysis from the nascent operational manipulations of Leibniz and John Bernoulli to the mature formulation of the Polish school. The work of A. Beer and C. Neumann in the nineteenth century on the Dirichlet problem led to Fredholm's work on integral equations. Although Hilbert made substantial progress and revealed the algebraic side of Fredholm and spectral theory, his work did not have the ``feel'' of modern functional analysis. It was Riesz, Schmidt, Volterra and Fréchet who developed the ideas of space, metric and functional, albeit in the context of specific concrete structures. Complete abstraction was achieved in the work of Banach and Hahn, who formulated the fundamental theorems of the field.