20. Function spaces.
We know that the notion of an arbitrary function did not emerge much before the beginning of the XIXth century. All the more so the idea of studying in a general way sets of functions, and furnishing them with a topological structure, did not appear before Riemann (see p. 141) and does not begin to be put to use before the end of the XIXth century.
However, the notion of convergence of a sequence of numerical functions was used in a more or less conscious way since the beginnings of the infinitesimal Calculus. But it was only a matter of simple convergence and it could not be otherwise before the notions of convergent series and of continuous functions had been defined accurately by Bolzano and Cauchy. This latter did not see at the first attempt the distinction between simple convergence and uniform convergence, and believed it possible to prove that every convergent series of continuous functions has as sum a continuous function ([56 a], (2), vol. III, p. 120). The error was almost immediately uncovered by Abel, who proved at the same time that every entire series is continuous in the interior of its convergence interval, by means of the argument that has become a classic which uses essentially, in this particular case, the idea of uniform convergence ([1], v. I, pp. 223-224). It only remained to disengage this latter in a general way, which was done independently by Stokes and Seidel in 1847-48, and by Cauchy himself in 1853 ([56 a], (1), vol. XII, p. 30).
Under the influence of Weierstrass and of Riemann, the systematic study of the notion of uniform convergence and of connected questions is developed in the last third of the XIXth century by the German school (Hankel, du Bois-Reymond) and especially the Italian school: Dini and Arzelà specify the necessary conditions for the limit of a sequence of continuous functions to be continuous, whereas Ascoli introduces the fundamental notion of equicontinuity and proves the theorem characterising compact sets of continuous functions [10] (a theorem popularised later by Montel in his theory of "normal families", which are none other than relatively compact sets of analytic functions).
In a work dated 1841, but published only in 1894 ([329 a], v. I, pp. 67-74), Weierstrass uses with perfect clarity the notion of uniform convergence (to which he gives this name for the first time) for power series of one or several complex variables.
Weierstrass himself discovered on the other hand ([329 a], v. III, pp. 1-37) the possibility of approximating uniformly by polynomials a continuous numerical function of one or several real variables on a bounded set, a result that aroused immediately a lively interest, and led to numerous "quantitative" studies that remain outside the point of view we follow at present.
The modern contribution to these questions consisted above all in giving them the whole range of which they are capable, by bringing them into play for functions whose set of definition and set of values are no longer restricted to or to spaces of finite dimension, and in placing them thus in their natural framework, with the help of general topological concepts. In particular, the theorem of Weierstrass, which had already shown itself to be a tool of the first order in classical Analysis, could, in these latter years, be extended to much more general cases by M. H. Stone: developing an idea introduced by H. Lebesgue (in a proof of the theorem of Weierstrass), he highlighted the important role played in the approximation of continuous numerical functions by lattices (approximation by "lattice polynomials"), and showed on the other hand how the generalised theorem of Weierstrass implies immediately a whole series of analogous approximation theorems, that are thus grouped in a much more coherent way [301 c].