Résumé En 1887 Volterra est lancé dans la recherche d'une vision générale pour l'analyse qui va prendre plusieurs formes. Ses travaux les plus connus le mènent ainsi à définir les fonctionnelles, ou plus précisément à développer un calcul différentiel et intégral pour des « fonctions qui dépendent d'autres fonctions » ou des « fonctions de lignes ». Pourtant les efforts de Volterra pour définir un contexte général pour certains problèmes d'analyse vont aussi l'amener à étendre les notions de dérivation et d'intégration aux substitutions—matrices dont les coefficients sont des fonctions—qui ont un rôle important dans l'étude des équations différentielles linéaires. Dans un mémoire intitulé Sui fondamenti della teoria delle equazioni differenziali lineari Volterra établit un calcul différentiel et intégral pour les substitutions. Ce travail, qui permet de penser les équations différentielles linéaires grâce à deux opérations sur les substitutions—dérivation et intégration, permet aussi d'analyser la stratégie de progression mise en œuvre par le mathématicien italien dans sa recherche d'une analyse généralisée dès le début de sa carrière. Nous examinons les processus de sélection et de réorganisation qui ont permis à Volterra de transposer une théorie bien établie pour les fonctions ordinaires à un cadre adapté aux substitutions. Nous mettons ainsi au jour une dynamique de progression vers le général révélant les éléments sur lesquels s'appuie sa pensée et les motifs qui l'animent. Loin d'être anecdotique, ce texte qui ne résout pourtant aucune conjecture, permet de voir une cohérence dans la manière de progresser de Volterra, et éclaire son rôle dans la recherche d'une analyse générale qui deviendra petit-à-petit l'analyse fonctionnelle du 20e siècle. In 1887 Volterra was in search of a general vision for analysis. His most famous work led him to the definition of functionals, or more precisely to develop a differential and integral calculus for “functions that depend on other functions” or “line functions.” However, Volterra's efforts to define a general context for certain analytical problems would also lead him to extend the notions of derivation and integration to substitutions—matrices whose coefficients are functions—which have an important role in the study of differential linear equations. In a memoir entitled Sui fondamenti della teoria delle equazioni differenziali lineari Volterra establishes a differential and integral calculus for substitutions. This work, which allows one to think of linear differential equations through two operations on substitutions—derivation and integration, also makes it possible to analyse the progression strategy implemented by the Italian mathematician in his search for a generalized analysis from the beginning of his career. We are examining the selection and reorganization processes that have enabled Volterra to transpose a well-established theory for ordinary functions to a framework adapted to substitutions. We thus reveal a dynamic of progress towards generality, and explore the elements on which his thoughts are based. Far from being an anecdotal, this text, which does not solve any conjecture, allows us to see a coherence in Volterra's way of progressing, and clarifies his role in the search for an analysis which would gradually become the functional analysis of the 20th century.
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