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Paris, Gauthier-Villars, 1922.

8vo (255x166 mm), vi-442 pages. binding : Original printed wrappers. Stains on covers.

Paul Lévy (1886-1971) French mathematician is among the founders of modern probability theory. In 1920, he was appointed professor of analysis at the École Polytechnique.

references: Le Cam [Paul Levy (1886-1971). in Sixth Berkeley Symposium p. xvi : "Paul Levy's early work, represented for instance by his doctoral thesis in 1911 and culminating in the 1922 volume on Analyse Fonctionnelle, revolves around the extension to an infinity of dimensions of the classical theorems relative to first and second order partial differential equations. This was influenced by Volterra's style of study of "functions of lines" and Hadamard's questions concerning the manner in which Green's functions depend on the contours of the domain. Paul Levy was able to extend many of the results relative to first order partial differential equations, but the second order equations led to problems of a very different nature. The study of the Laplacian in Hilbert space led both Giteaux and Levy to introduce independently the idea of mean values taken on balls or other convex subsets of Hilbert space. Giteaux, who was killed at the beginning of World War I, left unpublished manuscripts. Hadamard gave Levy the task of rewriting them for publication. Some of the unsolved questions raised by these papers were combined with Levy's own ideas and at least partially solved in the 1922 volume. This work of Levy is not well known today, perhaps because functional analysis took around that time a rather more abstract direction with

the introduction of Banach spaces and the ensuing emphasis on the general theory of linear operations, and perhaps because the extension of the Laplacian to Hilbert space by Levy and Giteaux turned out to be different from that needed in quantum field theory. The rather deep aspects of the geometry of Hilbert space studied by Levy still remain to be fully explored."].

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