By the end of the 19th century, mathematical analysis had become a rigorous edifice built on solid foundations. In particular, the concept of a limit, which had long proved a stumbling block for mathematicians, had been precisely defined thanks to Augustin Louis Cauchy (1789–1857; see Augustin Louis Cauchy, Tangente Hors-série 91, 2024), but above all to the German mathematician Karl Weierstrass (1815–1897). This had been made possible by constructing the real numbers, which until then had seemed to arise from intuition. Recall that the whole of analysis rests on the concept of a limit, particularly differentiation and integration. The subject therefore seemed capable of advancing only through minor additions. Yet, during that same century, several developments already held the seeds of a revolution: abstract structures and the beginnings of functional analysis.
Signs of things to come ------------------------
Following Évariste Galois (1811–1832), who laid the foundations of group theory shortly before dying in a duel in 1832 (Les groupes, Bibliothèque Tangente 80, 2023; Évariste Galois, Bibliothèque Tangente 82, 2023), several mathematicians—including the Englishman Arthur Cayley (1821–1895), the German Leopold Kronecker (1823–1891) and the Frenchman Camille Jordan (1838–1922)—identified several common axioms satisfied by different sets equipped with an operation. For example, addition on sets of numbers is associative: (a + b) + c = a + (b + c). But the same property also holds for permutations and for the composition of functions. After some trial and error, a consensus emerged: a set equipped with an associative operation, possessing an identity element and in which every element has an inverse, would be called a "group." Meanwhile, the German Hermann Grassmann (1809–1877) gave an abstract definition of what we now call a "vector space," intended to represent both the plane and the space in which we live. Inspired by this rather abstruse work, but keenly aware of its importance, the Italian Giuseppe Peano (1858–1932), also known for his formalization of the natural numbers, published Calcolo geometrico in 1888. In this book, he defined affine and vector spaces axiomatically. Set theory, then being developed by the German Georg Cantor (1845–1918), proved the ideal framework for elaborating these concepts.
At the same time, mathematicians began studying the convergence of sequences of functions.
Among them, several Italian mathematicians, including Giulio Ascoli (1843–1896) and Cesare Arzelà (1847–1912), proved results on the limits of such sequences under various assumptions. If Stefan Banach is to be believed: "It was Volterra who introduced functions of functions—that is, functions whose arguments and values are themselves functions." Vito Volterra (1860–1940) was also Italian and, outside mathematics, a fierce opponent of fascism.