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\noindent{\bf Maurice FR\'ECHET}\\
b. 10 September 1878 - d. 4 June 1973\\
\noindent
{\bf Summary}. Fr\'echet, one of the founders of modern analysis, also made
various original contributions to the probability calculus. He played an
important role in the organization of a European scientific community
in the area of probability and statistics.
\vspace{.5cm}
Ren\'e Maurice Fr\'echet was born at Maligny in the d\'epartement of the
Yonne, into a Protestant family, which moved to Paris soon after.
As a pupil at the Lyc\'ee Buffon,
M. Fr\'echet was taught by Jacques Hadamard (1865-1963), before the
latter left Paris for Bordeaux in 1894. In 1900, he was admitted
to the \'Ecole Normale Sup\'erieure, the most selective of Parisian
Institutions of higher learning. In 1906, he defended one of the
most dazzling French theses in mathematics of his time. In it,
Fr\'echet defined the concept of an abstract metric space, and
within this framework demonstrated the validity of Weierstrass's
theorem. Fr\'echet's thesis marks the beginnings of analysis on
appropriately structured abstract spaces, one of the
principal new directions in analysis of this century
(Arboleda, 1980; Taylor, 1982-1987).
Throughout his life, Fr\'echet retained his taste
for the most general theories and concepts, and some of those which he
introduced were real strokes of genius. Sometimes reproached
for not having proved truly ``difficult theorems", he
is often credited only with introducing new concepts, generalizing and
resetting problems in a broader new framework within which he could
prove ``easy theorems" enunciated for his own interest. It is, however, to
Fr\'echet in particular that we owe the first theory
of integration with respect to an ``abstract measure", in 1915.
Kolmogorov was to make use of this to axiomatize probability theory,
citing ``his master Fr\'echet" explicitly as his source. Whatever one's
views, Fr\'echet's scientific activity over a period of 60 years
was considerable, amounting to over 300 publications covering all
areas of pure and applied mathematics, and including about ten
important books.
After a period as a schoolteacher at the Besan\c{c}on (in 1907) and
Nantes (in 1908) Lyc\'ees, he was appointed lecturer
in the Faculty of Sciences at Rennes. Then in 1910, he became Professor in
the Faculty of Sciences at Poitiers, replacing Henri Lebesgue, who
had been appointed to Paris. Both appointments were due to the
direct intervention of Borel (q.v.). After the First World War, during
which he served as an Anglo-French interpreter, Fr\'echet was
appointed Professor in the Faculty of Sciences at the newly
liberated City of Strasbourg, showcase of French science.
Actuarial studies and statistics had been taught at Strasbourg for a
long time, within the framework of the German university, associated with names
such as Lexis (q.v) and von Bortkiewicz (q.v.). It was
here that Fr\'echet, conscious of his national and international
role, and meticulous in the discharge of his duties,
began to teach applied mathematics, statistics, actuarial studies, and
nomography. His first statistical papers date back to these
years. It was entirely to be expected that Fr\'echet should be
called to Paris in 1928, when the Institut Henri Poincar\'e (IHP)
was created, to develop the teaching of
probability under the leadership of Borel, its director and the
holder of the Chair in the Calculus of Probabilities and
Mathematical Physics at the Sorbonne. It should be noted that
Borel held the directorship of the IHP from 1928 until his death in
1956. Apart from Borel,
Fr\'echet was at the time the only French university academic of
international renown who
was interested in recent developments in
probability and statistics. Paul L\'evy who had just published a
remarkable book on the subject in 1925 (L\'evy, 1925) was a
Professor at the \'Ecole Polytechnique and never held a university
position.
Fr\'echet was first appointed lecturer in
probability at the Sorbonne's Rockefeller Foundation, and then from
the end of 1928 as Professor (without a Chair), was promoted
to the tenured Chair of General Mathematics in 1933 and to
the Chair of Differential and Integral Calculus in 1935. Finally, at the
start of 1941, he succeeded Borel in the Chair of the Calculus of
Probabilities and Mathematical Physics until his retirement in
1949. In 1928 (and at least until 1935) Fr\'echet was also put in charge of
lectures at the \'Ecole Normale Sup\'erieure. It was in this
capacity, with Borel's blessing, that he directed a sizeable number
of young mathematicians towards research in probability, in
particular Doeblin, Fortet, Lo\`eve, Ville, and others.
As soon as he reached Paris in 1928 Fr\'echet directed his research
towards the new ``theory of chain events", namely Markov (q.v.) chains. and
published the first mathematical synthesis on this topic in
1938. It was also Fr\'echet who initiated the study of ``random elements"
taking values in the most general spaces.
In the statistical field, the usual evaluation of his works was not
always enthusiastic, to which following severe comment of Harald
Cram\'er\footnote{``Half a century with probability theory: some
personal recollections", {\it The Annals of Probability}, {\bf 4}(1976),
509-546 (see p. 528).} attests:
``In early years Fr\'echet had been an outstanding mathematician,
doing pathbreaking work in functional analysis. He had taken up
probabilistic work at a fairly advanced age, and I am bound to say
that his work in this field did not seem very impressive to me."
\noindent Nevertheless, Fr\'echet was the author of several interesting
statistical papers, among them one on the one-dimensional Cram\'er-Rao
inequality, some contributions in econometrics, and in spatial
statistics.
His statistical researches came from a critical reflection on
the theory of errors. In a series of lectures, papers and
discussions, particularly with Paul L\'evy, between the
two World Wars, Fr\'echet mainly attacked the excessive
hegemony of the Gaussian
distribution in all areas of application of the theory of
errors, and provided alternative solutions. He demonstrated
the usefulness
of the Laplace probability density, $(exp - |x|), -\infty