Maurice Fréchet

How to use this course

Every student of analysis uses Maurice Fréchet’s ideas, usually without his name. The metric space, the abstract notion of compactness, and the idea that a function can be a single point of a space all come from his 1906 thesis. His name survives in the Fréchet derivative, Fréchet spaces, the Fréchet distance between curves, the Fréchet distribution of extreme values and the Fréchet bounds of probability. This course follows his long life, from a village in Burgundy to the Paris Academy of Sciences, which elected him at 78, and works through the main threads of his mathematics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 08 follow his work in analysis and topology (1904–1928); Lessons 09 and 10 follow his turn to probability and statistics (1927–1951).

Timeline

Maligny, Paris and Hadamard, 1878–1900

René Maurice Fréchet was born on 2 September 1878 in Maligny, a village in the Yonne department of Burgundy, to Jacques and Zoé Fréchet. The family was Protestant, a small minority in the France of the Third Republic.

In Paris he attended the Lycée Buffon. His mathematics teacher there, from 1890 to 1893, was Jacques Hadamard, then a young man who had just finished his own thesis and would soon prove the prime number theorem. Hadamard recognized his ability and encouraged him. The connection lasted for decades. Hadamard later supervised Fréchet’s doctorate, and Fréchet’s early research grew directly out of Hadamard’s work on functionals, the functions whose arguments are themselves functions.

In 1900 Fréchet entered the École Normale Supérieure, the elite school that trained French academics and teachers. Its students were close to the new theories of sets and of real functions then coming out of Paris: Émile Borel’s measure, René Baire’s classification of functions, and Henri Lebesgue’s integral, published in 1902.

The École Normale and the thesis, 1900–1906

Fréchet passed the agrégation in 1903 and in the same year published four research papers. He also worked closely with Borel. In 1905 he wrote up Borel’s lectures as a book, Leçons sur les fonctions de variables réelles et les développements en séries de polynômes.

His thesis, Sur quelques points du calcul fonctionnel, was submitted on 2 April 1906 under Hadamard’s supervision. Its starting point was a question of Hadamard and of Vito Volterra: how to extend the calculus of functions of numbers to functions of curves, surfaces and other functions. Fréchet’s answer was to stop asking what the points of a space are. He considered an arbitrary set with a notion of limit, or with a distance he called the écart, and asked which theorems of analysis survive.

The thesis introduced what Felix Hausdorff named, in 1914, the metric space. It also introduced the abstract notion of compactness, and it studied concrete spaces of functions, curves and sequences in the new framework. Lessons 04 and 05 present its main results. Few readers understood at first how much the abstraction would simplify analysis; within thirty years it was the standard language.

Lycées, Poitiers and the war, 1907–1919

Like many French mathematicians of his generation, Fréchet began his career in secondary schools. He taught at the lycée in Besançon in 1907–1908 and at the lycée in Nantes in 1908–1909. In 1908 he married Suzanne Carrive. In 1910 he became professor of mechanics at the Faculty of Science in Poitiers.

These were productive years. In 1907 he and the Hungarian Frigyes Riesz found, in notes published side by side in the Comptes rendus, the representation of continuous linear functionals on square-integrable functions (Lesson 06). In 1910 he showed that every separable metric space sits isometrically inside one fixed space of bounded sequences. In 1911 and 1912 he wrote on the general notion of differential (Lesson 07).

On 4 August 1914 he was mobilized. Because of his languages he was attached to the British Army as an interpreter, and he spent about two and a half years at or near the front. He kept writing mathematics when he could.

Metric spaces and compactness

For the general reader. To talk about nearness, you need a way to measure distance. Before Fréchet, distance meant the distance between points of a line, plane or space. He noticed that most of analysis uses only three properties of distance, so any set with a distance satisfying them can be studied the same way, whether its points are numbers, curves or functions.

Fréchet’s thesis called these sets “classes (E)”, the letter for écart. His main examples were \(\mathbb R^n\); the continuous functions on \([a,b]\) with \(d(f,g)=\max|f-g|\); the space of all real sequences with

in which convergence is coordinatewise convergence; and a space of curves (Lesson 05).

Compactness

Fréchet called a set compact if every infinite subset has a limit point, and extremal if it is compact and closed. In modern terms, an extremal set is a sequentially compact set. The name expressed the main theorem:

Proof sketch. If \(K\) is sequentially compact and \(f\) is continuous, take \(x_n\) with \(f(x_n)\to\sup f\) (possibly \(+\infty\)); a convergent subsequence \(x_{n_k}\to x\) gives \(f(x)=\sup f\), finite. Conversely, if a sequence \((x_n)\) of distinct points of \(K\) has no convergent subsequence, the set \(S=\{x_n\}\) is closed in \(K\) and every subset of it is closed. The function \(x_n\mapsto n\) is continuous on \(S\), and by the Tietze extension theorem (metric spaces are normal) it extends to a continuous unbounded function on \(K\).

In \(\mathbb R^n\) this is the Bolzano–Weierstrass theorem. Fréchet’s point was that the same statement, with the same proof, holds in spaces of functions, once one knows which sets are compact there.

Spaces of functions and curves

For the general reader. Once functions are points, one can ask which collections of functions are “small” in the sense of compactness. Fréchet recognized the theorem of Cesare Arzelà and Giulio Ascoli as an answer. He also defined a distance between curves that respects the order in which they are traced, which computational geometry still uses to compare GPS tracks and handwriting.

The Fréchet distance

For curves \(P,Q:[0,1]\to\mathbb R^n\), Fréchet defined in 1906

where \(\alpha,\beta\) range over continuous non-decreasing surjections of \([0,1]\) onto itself. A walker follows \(P\), a dog follows \(Q\), neither goes backwards, and \(d_F\) is the shortest leash that suffices. Two parametrizations of the same oriented curve are at distance 0, so \(d_F\) is a metric on curves up to reparametrization. In 1995 Helmut Alt and Michael Godau gave a polynomial-time algorithm for polygonal curves.

A universal space

Let \(\{x_n\}\) be dense in \(E\) and fix \(x_0\). Map \(x\mapsto\Phi(x)=\big(d(x,x_n)-d(x_0,x_n)\big)_{n\ge1}\). By the triangle inequality each coordinate is bounded by \(d(x,x_0)\), and \(|\Phi(x)_n-\Phi(y)_n|\le d(x,y)\). Choosing \(x_n\) close to \(x\) makes \(|\Phi(x)_n-\Phi(y)_n|\) close to \(d(x,y)\), so the sup equals \(d(x,y)\).

Linear functionals and Riesz–Fréchet

For the general reader. A linear functional assigns a number to each function, linearly. Integration against a fixed weight is the basic example. In 1907 Fréchet and Riesz proved that, for the square-integrable functions of Lebesgue’s new theory, there are no other continuous examples. The result is now the starting point of the theory of Hilbert spaces.

In 1903 Hadamard had represented continuous linear functionals on \(C[a,b]\) as limits of integrals, and Fréchet developed this in a series of papers Sur les opérations linéaires (1904–1907). Riesz found the definitive form for \(C[a,b]\) in 1909 with Stieltjes integrals. For \(L^2\) the answer is simpler.

Proof. If \(\Lambda=0\), take \(g=0\). Otherwise \(N=\ker\Lambda\) is a closed proper subspace, so there is a unit vector \(u\perp N\) (by orthogonal projection, which needs completeness). For any \(f\), \(\Lambda(f)u-\Lambda(u)f\in N\), so \(\langle\Lambda(f)u-\Lambda(u)f,u\rangle=0\), that is, \(\Lambda(f)=\langle f,\Lambda(u)u\rangle\). Set \(g=\Lambda(u)u\). Uniqueness: if \(\langle f,g-g'\rangle=0\) for all \(f\), take \(f=g-g'\). The norm identity follows from Cauchy–Schwarz and \(\Lambda(g)=\|g\|^2\).

Fréchet spaces

The sequence space of Lesson 04 is not normed, but it is a complete metric vector space whose topology comes from the countable family of seminorms \(p_n(x)=|x_n|\). Stefan Banach called complete metrizable topological vector spaces “spaces (F)” in Fréchet’s honour in 1932. Today a Fréchet space is usually taken to be also locally convex: its topology is given by a translation-invariant complete metric built from countably many seminorms. Examples include \(C^\infty(\mathbb R)\), the holomorphic functions on a disc with uniform convergence on compact sets, and the Schwartz space of distribution theory. The open mapping and closed graph theorems hold for Fréchet spaces.

The Fréchet derivative

For the general reader. The derivative of a function at a point is the best linear approximation to the function near that point. Fréchet took this sentence as the definition, which works whenever the inputs and outputs live in spaces where “linear” and “small” make sense. It is now how every analyst defines differentiability.

Fréchet gave the definition in notes of 1911 and a paper of 1912, and developed it for general spaces in a 1925 paper in the Annales de l’École Normale Supérieure. In finite dimensions it is the total derivative already found in Karl Weierstrass’s lectures and in textbooks of Otto Stolz and W. H. Young; Fréchet’s contribution was to see it as the right definition in infinite dimensions.

The weaker Gateaux derivative asks only that \(\frac{d}{dt}F(x+th)\big|_{t=0}\) exist for each direction \(h\) and be linear in \(h\). A Fréchet differentiable map is Gateaux differentiable with the same derivative; the converse fails, because Fréchet’s definition demands the limit uniformly over directions.

The proof is the usual contraction-mapping argument applied to \(x\mapsto x-DF(x_0)^{-1}(F(x)-y)\); completeness of \(X\) is exactly what it needs. Through this theorem, Fréchet’s derivative is the basic tool for nonlinear partial differential equations.

Abstract spaces without a distance

For the general reader. Some important kinds of convergence, such as pointwise convergence of functions, cannot be described by any distance. Fréchet looked for the weakest structure on which analysis still works, and his search helped produce general topology. His 1928 book Les espaces abstraits surveyed the field.

Closed sets are those containing the limits of their convergent sequences. Later authors added a fourth condition (if every subsequence has a further subsequence converging to \(x\), the sequence converges to \(x\)) to bring sequential convergence closer to topology. Fréchet also studied spaces defined by neighbourhoods, and Hausdorff’s 1914 axioms for topological spaces built partly on his work.

A topological space in which every single point is closed is now called a \(T_1\) space; older texts call it accessible, or a Fréchet space in a second sense. A Fréchet–Urysohn space is one where the closure of any set is obtained by adding limits of sequences from it. Metric spaces are Fréchet–Urysohn; many spaces of analysis are not.

Exercise 8.1 gives the proof.

The law of the largest value

For the general reader. How high will the highest flood of the next century be? Questions like this concern the largest of many random values. In 1927 Fréchet found one of the three possible limiting laws for such maxima, the one with a heavy tail, where rare values are much larger than typical ones. It is used today for floods, insurance losses and financial crashes.

Let \(X_1,X_2,\dots\) be independent with common distribution function \(F\), and \(M_n=\max(X_1,\dots,X_n)\). Then \(P(M_n\le x)=F(x)^n\). One looks for constants \(a_n>0,b_n\) such that \((M_n-b_n)/a_n\) has a non-degenerate limit \(G\).

Fréchet’s paper appeared in the Annales de la Société Polonaise de Mathématique in 1927. Ronald Fisher and Leonard Tippett found all three forms in 1928, and Boris Gnedenko proved in 1943 that there are no others and gave the domains of attraction. The three families are now combined as the generalized extreme value distribution.

Dependence, bounds and estimation

For the general reader. If you know the chance of rain and the chance of wind, what can you say about the chance of both? Without knowing how the two are linked, only a range. Fréchet found the exact range, for two events and for whole distributions. He also proved one of the first lower bounds on how accurately a parameter can be estimated from data.

Wassily Hoeffding found these bounds in 1940 in a paper that went largely unread outside Germany; Fréchet found them independently in 1951. The upper bound is attained by comonotone pairs \((F^{-1}(U),G^{-1}(U))\), the lower by countermonotone pairs \((F^{-1}(U),G^{-1}(1-U))\), with \(U\) uniform on \([0,1]\). In copula language, every copula lies between \(W(u,v)=\max(u+v-1,0)\) and \(M(u,v)=\min(u,v)\).

The information inequality (1943)

The bound is usually named after Calyampudi Radhakrishna Rao and Harald Cramér. Fréchet’s 1943 paper and Georges Darmois’s of 1945 reached it independently, as did Alexander Aitken and Harold Silverstone in 1942.

Proof. Let \(S=\partial_\theta\log f(X;\theta)\). Then \(\mathbb E S=\int\partial_\theta f=0\). Differentiating \(\int T f=\theta\) gives \(\mathbb E[TS]=1\), so \(\operatorname{Cov}(T,S)=1\). Cauchy–Schwarz: \(1\le\operatorname{Var}T\cdot\operatorname{Var}S=\operatorname{Var}T\cdot I(\theta)\).

Markov chains

The second volume of Fréchet’s Recherches théoriques modernes sur la théorie des probabilités (1938) gave a systematic treatment of Markov chains with finitely many states, based on the matrix of transition probabilities. His student Wolfgang Doeblin, in a thesis of 1938, extended the theory to general state spaces with the coupling method and the condition now named after him. In the 1930s Fréchet also campaigned against careless uses of the correlation coefficient as a measure of dependence.

Strasbourg and Paris, 1919–1949

After the war Alsace returned to France, and the French state rebuilt the University of Strasbourg as a showcase of French science. In 1919 Fréchet became its professor of higher analysis and director of its Mathematics Institute. At Strasbourg he began the work in probability and statistics that occupied much of the rest of his career. His first doctoral students, Ben-Zion Linfield (1925) and Feliks Burdecki (1926), finished there.

In 1928 he moved to Paris, first as a Rockefeller Foundation lecturer and then as professor at the Sorbonne, while also lecturing at the École Normale Supérieure until 1935. He held successive chairs of general mathematics (1933), differential and integral calculus (1935) and, from 1941, the calculus of probabilities and mathematical physics. The same year, 1928, he published Les espaces abstraits and was an invited speaker at the International Congress of Mathematicians in Bologna. In 1929 the Polish Academy elected him a foreign member, a sign of his close ties to the Warsaw school around Wacław Sierpiński.

Paris in the 1930s was a centre of probability, and Fréchet’s seminar trained many of its leaders. His doctoral students included Kinjirô Kunugui (1930), Nachman Aronszajn and Đuro Kurepa (1935), António Monteiro (1936), Wolfgang Doeblin (1938), Robert Fortet and Jean Ville (1939), and Ky Fan and Michel Loève (1941). Ville introduced martingales into probability in his thesis; Loève later built probability at Berkeley; Ky Fan co-wrote with Fréchet a book on combinatorial topology (1946). Doeblin, mobilized in 1939, took his own life in 1940 rather than be captured, leaving a sealed envelope at the Academy whose contents, opened in 2000, anticipated parts of stochastic calculus.

Late recognition and legacy, 1949–1973

Fréchet retired from his chair in 1949 but kept publishing for two more decades, including the anthology Pages choisies d’analyse générale (1953) and the essays Les mathématiques et le concret (1955). Recognition in France came late. He failed in several elections to the Académie des sciences before he was elected in 1956, aged 78. Abroad he was a Fellow of the Royal Society of Edinburgh (1947) and a foreign member of the Royal Netherlands Academy of Arts and Sciences (1950).

He was a committed supporter of Esperanto, published scientific articles in it, and from 1950 to 1953 presided over the International Association of Esperantist Scientists. He died in Paris on 4 June 1973, aged 94.

Legacy

• Analysis: metric spaces, abstract compactness and function spaces are the setting of modern analysis; Banach and Hilbert spaces grew from his framework.
• Calculus in infinite dimensions: the Fréchet derivative is the standard definition, and Fréchet spaces are the setting for distributions and smooth function spaces.
• Topology: his abstract spaces fed directly into the axioms of general topology.
• Probability and statistics: the Fréchet extreme-value law, the Fréchet bounds of copula theory, the Fréchet distance between curves, and the Fréchet mean in metric spaces are used across statistics, finance and computational geometry.

Connections

People linked to Maurice Fréchet across this collection. A name without a link has no page here yet. See Maurice Fréchet in the network of all 337 lives →

Teachers and advisors

• Jacques Hadamard 1890-1906

Students

• André Blanc-Lapierre 1945
• António Monteiro 1936
• Ben-Zion Linfield 1925
• Christian Pauc 1939
• Jean Ville 1939
• Kinjirô Kunugui 1930
• Ky Fan 1941
• Michel Loève 1941
• Nachman Aronszajn 1935
• Robert Fortet 1939
• Wolfgang Doeblin 1938
• Đuro Kurepa 1935

Influenced by

• Henri Lebesgue
• Vito Volterra
• Émile Borel

Influenced

• Felix Hausdorff
• Stefan Banach

Colleagues and collaborators

• Norbert Wiener 1920
• Paul Lévy
• Wacław Sierpiński
• Émile Borel

Rivals

• Frigyes Riesz 1907

Also mentioned in this life

• Bernard Bolzano
• Augustin-Louis Cauchy
• Ronald Fisher
• David Hilbert
• Andrey Markov
• Thomas Joannes Stieltjes
• Karl Weierstrass

Fields

• functional analysis 11 lives
• general topology 3 lives
• probability theory 23 lives
• statistics 7 lives

On CoursesHub

• Functional Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, René Maurice Fréchet, MacTutor History of Mathematics.
2. Maurice Fréchet, Wikipedia.
3. (René) Maurice Fréchet, Mathematics Genealogy Project.
4. A. E. Taylor, A study of Maurice Fréchet I–III, Archive for History of Exact Sciences 27 (1982), 34 (1985), 37 (1987).

Mathematics

1. M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1–74.
2. M. Fréchet, Sur les ensembles de fonctions et les opérations linéaires, C. R. Acad. Sci. Paris 144 (1907); F. Riesz, Sur une espèce de géométrie analytique des systèmes de fonctions sommables, same volume.
3. M. Fréchet, Les dimensions d’un ensemble abstrait, Math. Ann. 68 (1910).
4. M. Fréchet, Sur la notion de différentielle, C. R. Acad. Sci. Paris 152 (1911); La notion de différentielle dans l’analyse générale, Ann. Sci. ENS 42 (1925).
5. M. Fréchet, Sur la loi de probabilité de l’écart maximum, Ann. Soc. Polon. Math. 6 (1927), 93.
6. M. Fréchet, Les espaces abstraits, Gauthier-Villars, 1928.
7. M. Fréchet, Généralisations du théorème des probabilités totales, Fund. Math. 25 (1935), 379–387.
8. M. Fréchet, Sur l’extension de certaines évaluations statistiques au cas de petits échantillons, Rev. Inst. Internat. Statist. 11 (1943).
9. M. Fréchet, Sur les tableaux de corrélation dont les marges sont données, Ann. Univ. Lyon, Sect. A, 9 (1951), 53–77.
10. R. A. Fisher and L. H. C. Tippett, Limiting forms of the frequency distribution of the largest or smallest member of a sample, Proc. Cambridge Philos. Soc. 24 (1928).
11. H. Alt and M. Godau, Computing the Fréchet distance between two polygonal curves, Int. J. Comput. Geom. Appl. 5 (1995).
12. Fréchet differential, Encyclopedia of Mathematics.