## René Maurice Fréchet, Hon.F.R.S.E.

### by F Smithies

Maurice Fréchet was born on 2 September 1878 at Maligny (Yonne). He studied at the école Normale Supérieure in Paris and obtained his doctorate in 1906; his supervisor was Jacques Hadamard. He occupied junior posts at Besançon, Nantes and Rennes, and was appointed to a chair at the University of Poitiers in 1910. He moved in 1919 to Strasbourg, and in 1929 to Paris, where he held a professorship until his retirement in 1949; he was made an honorary professor in 1951.He married Suzanne Carrive on 3 September 1908; they had four children. He died on 4 June 1973.

Fréchet's doctoral thesis

*(Rendiconti del Circolo Matematico di Palermo,*

**22,**1-74) initiated the development of general topology. The theory of sets of points in finite-dimensional Euclidean space had been developed by Georg Cantor, Camille Jordan, Giuseppe Peano, émile Borel and many others; Vito Volterra, Giulio Ascoli and others, including Hadamard, had begun to think of real-valued functions as points of a space. Fréchet's thesis unified these two approaches and provided an axiomatic framework for further developments.

He started by giving a set of axioms for the notion of the limit of a convergent sequence; on the basis of these he defined such fundamental topological notions as derived set, closed set, perfect set and interior point. He introduced the important concept of compactness, in the forms known today as relative sequential compactness and sequential compactness, and established the fundamental properties of (sequentially) compact sets, e.g. that a continuous function defined on such a set is bounded and attains its upper and lower bounds.

In a later part of his thesis Fréchet gave for the first time the now familiar axioms for a metric space, giving examples that included various function spaces and sequence spaces. He distinguished carefully between complete and incomplete metric spaces, and between separable and non-separable spaces.

In other papers of the same period Fréchet discussed the form and properties of continuous linear functionals on a number of function spaces, a line of investigation that had been initiated by Hadamard. In particular, in 1907 he (and Frédéric Riesz independently) proved what is now known as the Fréchet-Riesz theorem on the representation of linear functionals on the space of functions of integrable square; their two notes appeared in the same number of the

*Comptes Rendus hebdomadaires des séances de l'Académie des Sciences.*In 1908, simultaneously with Erhard Schmidt, he introduced the use of geometrical language in the investigation of the properties of Hilbert space.

Most of Fréchet's later contributions must be passed over with a brief mention of a few of the more important ones. Among these were his intrinsic definition of a polynomial in abstract linear spaces (1909), the Fréchet differential of a functional (1911; extended to general mappings in 1925) and his pioneer theory of integration in abstract spaces (1914).

About 1927 Fréchet began to take up serious work on the theory of probability; this culminated in his massive two-volume contribution (1937, 1938) to Borel's

*Traité du calcul des probabilités et de ses applications*under the general title

*Recherches théoriques modernes sur la théorie des probabilités.*This included much original work of his own on the convergence of sequences of random variables, in which he was one of the first to make systematic use of abstract measure theory, and on finite Markov processes, to the theory of which he made substantial contributions. From 1934 onwards he also did some important work on the asymptotic behaviour of iterated densities.

Fréchet continued as an active mathematician long after his official retirement, contributing to such subjects as generalised correlation coefficients, random variables with values in an abstract space, and the classification of linear sets of zero measure. He created and applied the concepts of asymptotically almost periodic functions and para-analytic functions (analogues of analytic functions mapping one hyper-complex number system into another). Almost his last publication (1965) was a 94-page article on the life and work of émile Borel.

Besides his treatise on probability theory, Fréchet wrote several other books; among these were

*L'équation de Fredholm*(an early account of integral equations, written in collaboration with H B Heywood, 1912) and

*Les espaces abstraits*(1928), in which he systematised much of his early work on general topology and functional analysis.

Fréchet was a keen supporter of the artificial international language Esperanto, and published a number of mathematical papers in it; he was President of the International Scientific Esperanto Association from 1951 to 1954.

He was elected to the French Academy of Sciences in 1956, taking the seat vacated by the death of Borel; he was also an Officer of the Legion of Honour. He was an honorary member of the scientific academies of Poland, The Netherlands and Spain. He was elected an Honorary Fellow of the Royal Society of Edinburgh in 1947.

Maurice Fréchet's RSE obituary by F Smithies appeared in

*Royal Society of Edinburgh Year Book 1975,*31-33.