The Laplacian acting on functions of finitely many variables appeared in the works of Pierre Laplace (1749–1827) in 1782. After nearly a century and a half, the infinite-dimensional Laplacian was defined. In 1922 Paul Lévy (1886–1971) introduced the Laplacian for functions defined on infinite-dimensional spaces.

The infinite-dimensional analysis inspired by the book of Lévy Leçons d'analyse fonctionnelle attracted the attention of many mathematicians. This attention was stimulated by the very interesting properties of the Lévy Laplacian (which often do not have finite-dimensional analogues) and its various applications.

In a work (published posthumously in 1919) Gâteaux gave the definition of the mean value of the functional over a Hilbert sphere, obtained the formula for computation of the mean value for the integral functionals and formulated and solved (without explicit definition of the Laplacian) the Dirichlet problem for a sphere in a Hilbert space of functions. In this work he called harmonic those functionals which coincide with their mean values.

In a note written in 1919 , which complements the work of Gâteaux, Lévy gave the explicit definition of the Laplacian and described some of its characteristic properties for the functions defined on a Hilbert function space.

In 1922, in his book and in another publication Lévy gave the definition of the Laplacian for functions defined on infinite-dimensional spaces and described its specific features.