Although he had published comparatively little, he left other work unpublished and Paul Levy prepared this work for publication in three papers "R Gateaux, Sur la notion d'intégrale dans le domaine fonctionnel et sur la théorie du potentiel, Bulletin de la Société Mathématique de France" (1919), "R Gateaux, Fonctions d'une infinitee de variables indépendantes, Bulletin de la Société Mathématique de France" (1919), and "R Gateaux, Sur diverses questions du calcul fonctionnel, Bulletin de la Société Mathématique de France" (1922). These were substantial pieces of work, the first paper being 23 pages long and the third being 36 pages. It is in these papers that the "Gateaux derivative" appears. This is defined on locally convex topological vector spaces and generalises the idea of a directional derivative from differential calculus. It has important applications in physics and, unlike other generalisations of the notion of derivative, it is not linear.
The paper Sur la notion d'intégrale dans le domaine fonctionnel et sur la théorie du potentiel has before it in the journal first a statement by Hadamard followed by a statement by Paul Levy. Hadamard wrote :-
The name of R Gateaux, killed at the beginning of the war (September 1914), is well known to all those who are interested in functional analysis. Five notes in 1913 and in 1914 in the Compes rendus of the Academy of Sciences and in the Rendiconti of the Accademia dei Lincei, contained results of such importance that the Academy of Sciences decided to award him the Francoeur Prize in 1916.We also quote an extract from what Paul Lévy wrote:-
But the discoveries which he obtained without having time to publish them, still greatly exceed the preceding ones in importance, and show what large expectation French science had the right to expect of their author. They are fortunately not lost. we found in the papers of Gateaux notes were entrusted to me and in the examination of which Paul Lévy, highly qualified for this study by his previous work, agreed to lend me his assistance.
In particular two memoirs: one on functions of infinitely many independent variables, the other on potential theory, were almost completely written, at least in a provisional way. we begin the publication of them here. Paul Lévy took on the rather considerable work of their development - all friends of science and of France will be unanimous in giving him their thankful recognition.
The memoir on the theory of integrals and potentials in functional analysis is, though last in date (June 1914), that which we publish first. It is, in my opinion, the one which is richest in results. It opens a new and important chapter in functional analysis and reading it makes it possible to realise what an immense loss the person of Gateaux is to science.Paul Lévy also wrote an introduction to Fonctions d'une infinitee de variables indépendantes which appears in the same issue of the Bulletin de la Société Mathématique de France immediately following the previous paper by Gateaux:-
The draft found in the papers is certainly not a definitive version. Also I tried to respect his thinking while, where I could, making simplifications ...
I have given, moreover, in a final note, some indication of another draft of the same memoir which was also found.
This memoir was found in the papers of R Gateaux, in two successive drafts, both of the two being dated March 1914. The second is perhaps the definitive draft and it would be possible to publish it without appreciable modification. But it was unfinished and the analytic theory of functions was published according to the first draft. Though this draft is less perfect, being certainly intended only for the author himself to use in the second draft, I had to make only one change of any importance. ...