Gateaux was born on at Vitry-le-François, Marne, 222 years after another mathematician, Abraham de Moivre, was born there. His father had a small saddlery and upholstery business, and his mother was a seamstress. He was schooled at Reims, and in 1907 entered the École normale supérieure on the rue d'Ulm. He was well regarded as one of the most promising mathematicians among his peers. During his time at ENS, Gateaux converted to Roman Catholicism.
Schoolteacher
In 1910, he sat the mathematics examination. He became a teacher at the lycée in Bar-le-Duc, Meuse in 1912, having completed his two years' military service. At the same time as he took the post at Bar-le-Duc, he started to work on his thesis about functional analysis, following the work of Vito Volterra and Jacques Hadamard, and its applications to potential theory. Even though it is unknown why Gateaux chose this subject, he may have been encouraged by Hadamard himself, who had just completed a course on the subject at the Collège de France. Among others, in 1911 Paul Lévy had undertaken a brilliant thesis on this type of question, and in 1912 Joseph Pérès, an alumnus of the ENS in the year before Gateaux, had left for Rome to work under Volterra.
Student in Rome
In 1913, Gateaux asked for, and was awarded, a bursary from the David Weill Foundation to go with him to Rome. Before leaving, he sent a letter to Borel and Volterra, on the subjects they had proposed he work on in Rome. Within it was the theme of integration of real functions in infinite-dimensional space. He stayed in Rome from October 1913, following Volterra's course and working hard. He published numerous notes in the Rendiconti dell'Accademia dei Lincei, and presented a seminar at the University of Rome. He returned to France in June 1914, intending to return in the September after being awarded a Commercy bursary for another year.
In August 1915, Hadamard started the process of awarding Gateaux a Prix de L'Académie des Sciences posthumously. In a letter to Émile Picard, he wrote: In 1916, Gateaux was awarded the Prix Francoeur. In 1918, Hadamard talked to Paul Lévy, who was responsible for a course on functional analysis at the Collège de France, about drafts Gateaux had left before his departure to the Front. He proposed that Lévy edit them for publication in the Bulletin de la Société Mathématique de France, in two parts. The most important discovery that Lévy found in Gateaux' papers was a draft theory for integrating functions in infinite dimensions. This work came to be of great importance to Lévy in writing his important work Leçons d'analyse fonctionnelle. When Lévy was talking withthe American mathematician Norbert Wiener in 1922, Wiener immediately saw that he could use Gateaux' definition to define his "differential space" and construct a measure of Brownian motion. In the foundation article that he published in 1923, Wiener paid homage to Gateaux and Lévy for producing les études les plus profondes sur l'intégration en dimension infinie.
