Mathematics Symposium, organised by the Department of Mathematical Sciences of CIÊNCIAS, featuring Laurent Mazliak (Sorbonne University, LPSM, Paris).
Abstract: Maurice Fréchet’s 1906 thesis, Sur quelques points du Calcul Fonctionnel, marked a turning point in modern analysis. Initially influenced by Hadamard, Borel, and Volterra, Fréchet moved beyond Volterra’s framework to propose a radically new approach: instead of repeating proofs for specifc domains, he introduced an abstract framework unifying results before applying them to functional spaces. He developed a general notion of limit in metric spaces, defined distance tructures, and introduced classes of topological spaces. His work encompassed key results such as the Arzelà–Ascoli theorem and promoted reasoning about highly abstract sets. Although sometimes viewed as a precursor to Bourbaki, Fréchet pursued abstraction mainly for methodological economy, using abstract spaces as tools rather than as a structural program. My presentation discusses the recent critical reissue of the thesis, prepared with Frédéric Jaëck, including a historical introduction and detailed commentary.