# Aryeh Dvoretzky

### Born: 3 May 1916 in Khorol, Ukraine

Died: 8 May 2008 in Jerusalem, Israel

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**Aryeh Dvoretzky**'s family were Jewish. Aryeh was born in Khorol, a town roughly half way between Kiev and Kharkov, and he spent the first six years of his life there before his family emigrated to Palestine in 1922. He attended the Hebrew Reali School in Haifa which had been founded just a few years earlier in protest against a decision to teach all exact sciences in German. The principal of the school was Dr Arthur Biram who developed the school greatly during the years that Dvoretzky was a pupil.

After graduating from the Hebrew Reali School, Dvoretzky entered the Hebrew University of Jerusalem. This university, which opened in 1925, had only awarded its first degrees a few years before Dvoretzky began his studies there. He was taught by, among others, Michael Fekete and Abraham Fraenkel. Fekete was a Hungarian mathematician who had emigrated to Palestine in 1928 and was promoted to a professor at the Hebrew University in the following year. Fraenkel, who joined the university in 1929, was its first dean of mathematics. Dvoretzky was awarded his Master's Degree by the Hebrew University in 1937 and continued to undertake research there for his doctorate advised by Fekete. He was awarded his doctorate in 1941 for his thesis

*Studies on general Dirichlet series*but before submitting this work he had already published a number papers in French, the first being

*Sur les singularités des fonctions analytiques*(1938). In fact his thesis consisted of five chapters, the first three of which had been published as three separate papers in

*Comptes Rendus*of the Academy of Sciences in Paris.

In 1940 Dvoretzky married Sarah Schneerson (1915-1972), who was also born in the Ukraine. Sarah had come with her family to Tel Aviv in 1924, and studied philosophy and classical languages at the Hebrew University in Jerusalem receiving her Master's Degree in 1938. She became a well-known translator of classical works into Hebrew. Aryeh and Sarah Dvoretzky had two children, a daughter Gina and a son Gideon who was killed on the Egyptian front in the Yom Kippur War of 1973. After the award of his doctorate in 1941, Dvoretzky was appointed to the teaching staff of the Hebrew University of Jerusalem. He was promoted to a full professorship in 1951, becoming the first graduate of the Hebrew University to be appointed to a chair in that university. He took a full part in the running of the University being the Dean of the Faculty of Sciences in 1955-1956 and Vice President of the Hebrew University from 1959 to 1961.

Dvoretzky made many research visits abroad, mainly to the United States but also to the Collège de France. He visited Columbia University, Purdue University, Stanford University, and the University of California at Berkeley as well as the Institute for Advanced Study in Princeton on two separate occasions. He first visit was during the academic years 1948-1950 and the second visit was in 1957-1958. These visits were particularly significant for in 1975 he founded the Institute for Advanced Studies of Jerusalem, modelled on the Institute for Advanced Study at Princeton. In March, 1976 Dvoretzky wrote about the new Institute:

He also wrote:-The Institute is similar in concept to several existing Institutes of Advanced Study, notably the Princeton Institute. An IAS in Israel will fulfil a long-acknowledged need for an appropriate setting to encourage scientific and academic leadership, along with promoting the highest standard of research. The proliferation of universities in Israel, along with the overall trend toward mass higher education, has heightened the need for an IAS here in Israel. The inspiration and achievement of these Institutes are essential to strengthening and advancing Israel's scientific and academic landscape.

In 1960, he became the head of Rafael, the Israeli weapons development authority. He later became the chief scientist at the Israel Ministry of Defense. There is an interesting account in [1] of Dvoretzky's advice to the Israeli Prime Minister Levi Eshkol concerning the Palestinian refugee problem after the Six Day War in 1967.The Institute for Advanced Studies of Jerusalem aims for excellence. In Jerusalem we must have either an outstanding Institute or none at all.

Let us now look briefly at Dvoretzky's mathematical contributions. Joseph Yahav writes in [4] about his early interest in analysis, then explains about his work on probability theory:-

We should say a little more about the Dvoretzky theorem. In 1950 Dvoretzky published a joint paper with Ambrose Rogers entitledDvoretzky's interest in probability started in the early1940s; we see his work with Theodore Motzkin in the Duke Mathematical Journal, vol.14, "A problem of arrangements," and in1946, the "Lectures on the Theory of Probability"(in Hebrew). From1950his publications were mainly in English. As mentioned, his main field of work was analysis and convexity theory. His best known fundamental result in this field is the Dvoretzky theorem, which was related by Vitali Milman to Paul Lévy's measure concentration phenomena and served as a starting point to modern Banach space theory. At the same time, Dvoretzky was producing work on probability. His work "On the Strong Stability of Sequence of Events" was published in the Annals of Mathematical Statistics, vol.20, in1950. At this time he was cooperating with Shizuo Kakutani and Paul Erdős, working on Brownian motion in n-space. Throughout the years, Dvoretzky cooperated with Erdős, Jacob Wolfowitz, Abraham Wald, Herbert Robbins and Y S Chow in producing elegant and fundamental work in probability theory.

*Absolute and Unconditional Convergence in Normed Linear Spaces*. In this paper they proved that, for a series of points in a Banach space, absolute convergence is equivalent to unconditional convergence if and only if the Banach space is finite-dimensional. This was an impressive paper since this result had been an open conjecture for twenty years. This result inspired Alexander Grothendieck who produced a number of different proofs. One such proof appeared in Grothendieck's 1956 paper

*Sur certaines classes de suites dans les espaces de Banach et le théorème de Dvoretzky-Rogers*which also contained a number of important conjectures. In 1959 Dvoretzky proved the conjecture by Grothendieck which today is known as Dvoretzky's theorem. Its proof, which depends on some intricate measure-theoretic arguments, is sketched in the paper

*A theorem on convex bodies and applications to Banach spaces*and discussed in greater detail in

*Some results on convex bodies and Banach spaces*(1960). Keith Ball, reviewing [3], writes:-

Milman [3] writes about the impact of Dvoretzky's theorem:-In1959A Dvoretzky proved that, for every positive e and every n, all normed spaces of sufficiently large dimension contain n-dimensional subspaces which are said to be within e of n-dimensional Euclidean space. ... Thus, all symmetric convex bodies of sufficiently large dimension have central sections, indistinguishable to the naked eye, from n-dimensional ellipsoids. Dvoretzky's theorem initiated an avalanche of work on finite-dimensional normed spaces, guided by the heuristic principle: "All convex bodies behave like ellipsoids, and ever more so as the dimension increases." This principle runs completely counter to one's initial experience. As the dimension n increases, cubes seem less and less like spheres. The ellipsoid of largest volume inside an n-dimensional cube is the obvious solid sphere, and occupies only a tiny fraction ... of the cube's volume. Nevertheless there are remarkable theorems embodying the heuristic principle and providing powerful tools for the study of high-dimensional geometry.

Among his other work we must emphasise his important contributions to Brownian motion, already mentioned above, much of which was done in collaboration with Shizuo Kakutani and Paul Erdős. They wrote several papers on this topic including:It soon became clear that an outstanding breakthrough in Geometric Functional Analysis had been achieved.

*Double Points of Paths of Brownian Motion in n-Space*(1950),

*Multiple points of paths of Brownian motion in the plane*(1954),

*Triple points of Brownian paths in 3-space*(1957),

*Points of multiplicity c of plane Brownian paths*(1958), and

*Nonincrease everywhere of the Brownian motion process*(1961). Later Dvoretzky wrote a number of single-author papers on Brownian motion including

*On the oscillation of the Brownian motion process*(1963) and

*Polygons on two-dimensional Brownian paths*(1986).

Not only did Dvoretzky make a major contribution to mathematical research at the Hebrew University but he also was an excellent teacher [4]:-

Among many doctoral students that he advised at the Hebrew University we mention three who went on to a highly successful academic career; Branko Grünbaum, who wrote the thesisHe was considered one of the best teachers in the Hebrew University's Math Department, inspiring many students to take probability as their field of research.

*On Some Properties of Minkowski Spaces*(1957), Joram Lindenstrauss, who wrote the thesis

*Extension of Compact Operators*(1962), and Aldo Joram Lazar, who wrote the thesis

*Spaces of Affine Functions on Simplexes*(1968).

Dvoretzky received many honours for his contributions. In particular we note that he was an invited speaker at the International Congress of Mathematicians held in Nice in 1970. He was awarded the Israel Prize for Mathematics in 1973. He was elected president of the Israel Academy of Sciences and Humanities in 1974, having been a founder member of the Academy, serving until 1980. Later he became the eighth president of the Weizmann Institute of Science serving from 1986 to 1989. In 1996 he was awarded an honorary doctorate from Tel Aviv University. In 2009, the Einstein Institute of Mathematics at the Hebrew University established an annual lecture series in memory of Dvoretzky.

**Article by:** *J J O'Connor* and *E F Robertson*

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