In the history of mathematics there are quite a few persons whom we prefer to recollect in pairs. Listed among them are Euclid and Diophant, Newton and Leibniz, Bolyai and Lobachevsky, Hilbert and Poincaré, as well as Bourbaki and Arnold. In this series we enroll Sobolev and Schwartz who are inseparable from one of the most brilliant discoveries of the twentieth century, the theory of generalized functions or distribution theory, providing a revolutionary new approach to partial differential equations.
The most vibrant and lasting achievements of mathematics reside in formulas and lists. There are pivotal distinctions between lists and formulas. The former deposit that which was open for us. The lists of platonic solids, elementary catastrophes, and finite simple groups are next of kin to the Almagest and herbaria. They are the objects of admiration, tremendous and awe-struck. The article of the craft of mathematics is a formula. Each formula enters into life as an instance of materialization of mathematical creativity. No formula serves only the purpose it was intended to. In part, any formula is reminiscent of household appliances, toys, or software. It is a very rare event that somebody reads the user's guide of a new TV set or the manual for running a new computer program. Usually everyone utilizes his or her new gadgets experimentally by pressing whatever keys and switches. In much the same way we handle formulas. We painstakingly “twist and turn” them, audaciously insert new parameters, willfully interpret symbols, and so on.
Mathematics is the craft of formulas and the art of calculus. If someone considers this claim as feeble and incomplete, to remind is in order that, logically speaking, set theory is just an instance of the first order predicate calculus.
Distribution theory has become the calculus of today. Of such a scale and scope is the scientific discovery by Sobolev and Schwartz.
Sobolev was born in St. Petersburg on October 6, 1908 in the family of Lev Aleksandrovich Sobolev, a solicitor. Sobolev's grandfather on his father's side descended from a family of Siberian Cossacks.
Sobolev was bereaved of his father in the early childhood and was raised by his mother Natal′ya Georgievna who was a highly-educated teacher of literature and history. His mother also had the second speciality: She graduated from a medical institute and worked as a tutor at the First Leningrad Medical Institute. She cultivated in Sobolev the decency, indefatigability, and endurance that characterized him as a scholar and personality.
Sobolev fulfilled the program of secondary school at home, revealing his great attraction to mathematics. During the Civil War he and his mother lived in Kharkov. When living there, he studied at the preparatory courses of a evening technical school for one semester. At the age of 15 he completed the obligatory programs of secondary school in mathematics, physics, chemistry, and other natural sciences, read the classical pieces of the Russian and world literature as well as many books on philosophy, medicine, and biology.
After the family had transferred from Kharkov to Petersburg in 1923, Sobolev entered the graduate class of and finished with honors in 1924, continuing his study at the First State Art School in the piano class. At the same year he entered the Faculty of Physics and Mathematics of Leningrad State University (LSU) and attended the lectures of Professors N. M. Günter, V. I. Smirnov, G. M. Fikhtengolts, and others. He made his diploma on the analytic solutions of a system of differential equations with two independent variables under the supervision of Günter.
Günter propounded the idea that the set functions are inevitable in abstracting the concept of solution to a differential equation. Günter's approach influenced the further train of thought of Sobolev.2
After graduation from LSU in 1929, Sobolev started his work at the Theoretical Department of the Leningrad Seismological Institute. In a close cooperation with Smirnov he then solved some fundamental problems of wave propagation. It was Smirnov whom Sobolev called his teacher alongside Günter up to his terminal days.
Since 1932 Sobolev worked at the Steklov Mathematical Institute in Leningrad; and since 1934, in Moscow. He continued the study of hyperbolic equations and proposed a new method for solving the Cauchy problem for a hyperbolic equation with variable coefficients. This method was based on a generalization of the Kirchhoff formula. Research into hyperbolic equations led Sobolev to revising the classical concept of a solution to a differential equation. The concept of a generalized or weak solution of a differential equation was considered earlier. However, it was exactly in the works by Sobolev that this concept was elaborated and applied systematically. Sobolev posed and solved the Cauchy problem in spaces of functionals, which was based on the revolutionary extension of the Eulerian concept of function and declared 1935 as the date of the birth of the theory of distributions.
Suggesting his definition of generalized derivative, Sobolev enriched mathematics with the spaces of functions whose weak derivatives are integrable to some power. These are now called Sobolev spaces.
The contributions of Sobolev brought him recognition in the USSR. In 1933 Sobolev was elected a corresponding member of the Academy of Sciences at the age of 24 years. In 1939 he became a full member of the Academy and remained the youngest academician for many years.
Inspired by military applications in the 1940s, Sobolev studying the system of differential equations describing small oscillations of a rotating fluid. He obtained the conditions for stability of a rotating body with a filled-in cavity which depend on the shape and parameters of the cavity. Moreover, he elaborated the cases in which the cavity is a cylinder or an ellipsoid of rotation. This research by Sobolev signposted another area of the general theory which concerns the Cauchy and boundary value problems for the equations and systems that are not solved with respect to higher time derivatives.
In the grievous years of the Second World War from 1941 to 1944 Sobolev occupied the position of the director of the Steklov Mathematical Institute.
Sobolev was one of the first scientists who foresaw the future of computational mathematics and cybernetics. From 1952 to 1960 he held the chair of the first national department of computational mathematics at Moscow State University. This department has played a key role in the development of this important area of the today's mathematics.
Addressing the problems of computational mathematics, Sobolev lavishly applied the apparatus of the modern sections of the theoretical core of mathematics. It is typical for him to pose the problems of computational mathematics within functional analysis. Winged are his words that “to conceive the theory of computations without Banach spaces is impossible just as trying to conceive it without computers.”
It is worthwhile to emphasize the great role in the uprise of cybernetics, genetics, and other new areas of research in this country which was played by the publications and speeches of Sobolev who valiantly defended the new trends in science from the ideologized obscurantism.
It is difficult to overrate the contribution of Sobolev to the design of the nuclear shield of this country. From the first stages of the atomic project of the USSR he was listed among the top officials of Laboratory No. 2 which was renamed for secrecy reasons into the Laboratory of Measuring Instruments (abbreviated as LIPAN in Russian). Now LIPAN lives as the Kurchatov Center. The main task of the joint work with I. K. Kikoin was the implementation of gaseous diffusive uranium enrichment for creation of a nuclear explosive device.
Sobolev administered and supervised various computational teams, studied the control of the industrial processes of isotope separation, struggled for the low costs of production and made decisions on many managerial and technological matters. For his contribution to the A-bomb project, Sobolev twice gained a Stalin Prize of the First Degree. In January of 1952 Sobolev was awarded with the highest title of the USSR: he was declared the Hero of the Socialist Labor for exceptional service to the state.
Sobolev's research was inseparable from his management in science. At the end of the 1950s M. A. Lavrent′ev, S. L. Sobolev, and S. A. Khristianovich came out with the initiative to organize a new big scientific center, the Siberian Division of the Academy of Sciences. For many scientists of the first enrolment to the Siberian Division it was the example of Sobolev, his authority in science, and the attraction of his personality that yielded the final argument in deciding to move to Novosibirsk.
The Siberian period of Sobolev's life in science was marked by the great achievements in the theory of cubature formulas. Approximate integration is one of the main problems in the theory of computations-the cost of computation of multidimensional integrals is extremely high. Optimizing the formulas of integration is understood now to be the problem of minimizing the norm of the error on some function space. Sobolev suggested new approaches to the problem and discovered marvelous classes of optimal cubature formulas.
Sobolev merits brought him many decorations and signs of distinction. In 1988 he was awarded the highest prize of the Russian Academy of Sciences, the Lomonosov Gold Medal.
Sobolev passed away in Moscow on January 3, 1989.
Schwartz was born in Paris on March 5, 1915 in the family of Anselme Schwartz, a surgeon. There were quite a few prominent persons among his next of kin. J. Hadamard was his granduncle. Many celebrities are listed in the lone of his mother's line Claire Debrés: Several Gaullist politicians belonged to the Debérs. In 1938 Schwartz married Marie-Hélèn Lévy, the daughter of the outstanding mathematician P. Lévy who was one of the forefathers of functional analysis. Marie-Hélèn had become a professional mathematician and gained the position of a full professor in 1963.
The munificent gift of Schwartz was revealed in his lecée years. He won the most prestigious competition for high school students, Concours Général in Latin. Schwartz was unsure about his future career, hovering between geometry and “classics” (Greek and Latin). It is curious that Hadamard had a low opinion of Schwartz mathematical plans, since the sixteen-years old Laurent did not know the Riemann zeta function. By a startling contrast, Schwartz was boosted to geometry by the pediatrician Robert Debré and one of his teachers of classics.
In 1934 Schwartz passed examinations to the École Normale Supérieure (ENS) after two years of preparation. He was admitted together with Gustave Choquet, a winner of the Concours Général in mathematics, and Marie-Hélèn, one of the first females in the ENS. The mathematical atmosphere of those years in the ENS was determined by É. Borel, É. Cartan, A. Denjoy, M. Fréchet, and P. Montel. The staff of the neighboring Collège de France included H. Lebesgue who delivered lectures and Hadamard who conducted seminars. It was in his student years that Schwartz had acquired his irretrievable and permanent love to probability theory which grew from conversations with his future father-in-law Lévy.
After graduation from the ENS Schwartz decided to be drafted in the compulsory military service for two years. He had to stay in the army in 1939-1940 in view of the war times. These years were especially hard for the young couple of the Schwartzes. It was unreasonable for Jews to stay in the occupied zone. The Schwartzes had to escape from the native north and manage to survive on some modest financial support that was offered in particular by Michelin, a world-renowned tire company. In 1941 Schwartz was in Toulouse for a short time and met H. Cartan and J. Delsarte who suggested that the young couple should move to Clermont-Ferrand, the place of temporary residence of the group of professors of Strasbourg University which had migrated from the German occupation. These were J. Dieudonné, Ch. Ehresmann, A. Lichnerowicz, and S. Mandelbrojt. In Clermont-Ferrand Schwartz completed his Ph. D. Thesis on approximation of a real function on the axis by sums of exponentials.
Unfortunately, the war had intervened into the mathematical fate of Schwartz. His family had to change places under false identities. Curiously, in the time of the invention of distributions in November of 1944 Schwartz used the identity of Selimartin. The basis of Schwartz's theory appeared in the Annales of the University of Grenoble in 1945 . Schwartz described the process of invention as “cerebral percolation.” After a year's stay in Grenoble, Schwartz acquired a position in Nansy, plunging to the center of “Bourbakism.” It is well known that N. Bourbaki resided in Nancago, a mixture of Nancy and Chicago. A. Weyl lived in Chicago, while Delsarte and Diedonné were in Nancy. Before long Schwartz was enrolled in the group of Bourbaki. In 1950 he was awarded with the Fields medal for distribution theory. His now-celebrated two-volume set Théorie des Distributiones was printed a short time later.
In 1952 Schwartz returned back to Paris and began lecturing in Sorbonne; and since 1959, in the École Politechnique in company with his father-in-law Lévy. Many celebrities were the direct students of Schwartz. Among them we list A. Grothendieck, J.-L. Lions, B. Malgrange, and A. Martineau.
Schwartz wrote: “To discover something in mathematics is to overcome an inhibition and a tradition. You cannot move forward if you are not subversive.” This statement is in good agreement with the very active and versatile public life of Schwartz. He joined Trotskists in his green years, protesting against the monstrosities of capitalism and Stalin's terror of the 1930s. Since then he had never agreed with anything that he viewed as violation of human rights, oppression, or injustice. He was very active in struggling against the American war in Vietnam and the Soviet invasion in Afghanistan. He fought for liberation of a few mathematicians that were persecuted for political reasons, among them Jose Luis Massera, Vaclav Benda, et al.
Schwartz was an outstanding lepidopterist and had collected more than 20 000 butterflies. It is not by chance that the butterflies are depicted on the soft covers of the second edition of his Théorie des Distributiones.
Schwartz passed away in Paris on June 4, 2002.
Distribution theory stems from the intention to apply the technologies of functional analysis to studying partial differential equations. Functional analysis rests on algebraization, geometrization, and socialization of analytical problems. By socialization we usually mean the inclusion of a particular problem in an appropriate class of its congeners. Socialization enables us to erase the “random features,”3 eliminating the difficulties of the insurmountable specifics of a particular problem. In the early 1930s the merits of functional analysis were already demonstrated in the area of integral equations. The time was ripe for the differential equations to be placed on the agenda.
It is worth observing that the contemplations about the nature of integration and differentiation underlie most of the theories of the present-day functional analysis. This is no wonder at all in view of the key roles of these remarkable linear operations. Everyone knows that integration possess a few more attractive features than differentiation: the integral is monotone and raises smoothness. The derivations completely lacks these nice properties. Everyone knows as well that the classical derivative yields a closed yet unbounded operator (with respect to the natural uniform convergence topology that is induced by the Chebyshev sup-norm). The series of smooth functions cannot be differentiated termwise in general, which diminishes the scope of applications of analysis to differential equations.
There is practically no denying today that the concept of generalized derivative occupies a central place in distribution theory. Derivation is now treated as the operator that acts on the nonsmooth functions according to the same integral laws as the procedure of taking the classical derivative. It is exactly this approach that was pursued steadily by Sobolev. The new turnpike led to the stock of previously impossible differentiation formulas. It turned out that each distribution possesses derivatives of all orders, every series of distributions may be differentiated termwise however often, and many “traditionally divergent” Fourier series admit presentations by explicit formulas. Mathematics has acquired additional fantastic degrees of freedom, which makes immortal the name of Sobolev as a pioneer of the calculus of the twentieth century.
The detailed expositions of the new theory by Sobolev and Schwartz had appeared practically at the same time. In 1950 the first volume of Théorie des Distributiones was published in Paris, while Sobolev's book Applications of Functional Analysis in Mathematical Physics was printed in Leningrad. In 1962 the Siberian Division of the Academy of Sciences of the USSR reprinted the book, while in 1963 it was translated into English by the American Mathematical Society. The second edition of the Schwartz book was published in 1966, slightly enriched with a generalized version of the de Rham currents. Curiously, Schwartz left the historical overview practically the same as in the introduction to the first edition.
The new methods of distribution theory turned out so powerful as to enable mathematicians to write down, in explicit form, the general solution of an arbitrary partial differential equation with constant coefficients. In fact, everything reduces to existence of fundamental solutions; i.e. to the case of the Dirac delta-function on the right-hand side of the equation under consideration. The existence of these solutions was already established in 1953 and 1954 by B. Malgrange and L. Ehrenpreis independently of each other. However, it was only in 1994 that some formula for a fundamental solution was exhibited by H. König. Somewhat later N. Ortner and P. Wagner found a more elementary formula. Their main result is as follows:4
It stands to reason to inspect the structure of the formula which reveals the role of the distributional Fourier transform 𝔉 and the Schwartz space 𝒮′(ℝn) comprising tempered distributions.5
The existence of a fundamental solution of an arbitrary partial differential equation with constant coefficients is reverently called the Malgrange-Ehrenpreis Theorem. It is hard to overestimate this splendid achievement which remains one of the splendid triumphs of the abstract theory of topological vector spaces.
The road from solutions in distributions to standard solutions lies through Sobolev spaces. Study of the embeddings and traces of Sobolev spaces has become one of the main sections of the modern theory of real functions. Suffice it to mention S. M. Nikol′ski, O. V. Besov, G. Weiss, V. P. Il′in, and V. G. Mazya in order to conceive the greatness of this area of mathematical research. The titles of dozen books mention Sobolev spaces, which is far from typical in the present-day science.
The broad stratum of modern studies deals with applications of distributions in mathematical and theoretical physics, complex analysis, the theory of pseudodifferential operators, Tauberian theorems, and other sections of mathematics.
The physical sources of distribution theory as well as the ties of the latter with theoretical physics, are the topics of paramount importance. They require a special scrutiny that falls beyond the scope of this article.6 We will confine exposition to the concise historical comments by V. S. Vladimirov:7
J. Leray was one of the most prominent French mathematicians of the twentieth century. He was awarded with the Lomonosov Gold Medal together with Sobolev in 1988. Reviewing the contributions of Sobolev from 1930 to 1955 in the course of Sobolev's election to the Academy of Sciences of the Institute of France in 1967, J. Leray wrote:8
In sharp contrast with this appraisal, we cite F. Tréves who wrote in the memorial article about Schwartz in October 2003 as follows:10
F. Tréves should be honored for drifting aside from the practice of evaluating publications from what they lack when he wrote somewhat later about that which made the name of Schwartz immortal:12
The Tréves opinion coincides practically verbatim with the narration of Schwartz in his autobiography published firstly in 1997. Moreover, Schwartz had even remarked there about Sobolev that13
It is impossible to agree with these opinions. Rather strange is to read about the absence of any mention of the Dirac delta-function among the generalized functions of Sobolev, since δ obviously belongs to each of the spaces (Cmcomp)∗.
Disappointing is the total neglect of the classical treatise of Sobolev  which was a deskbook of many specialists in functional analysis and partial differential equations for decades.14 Finally, Schwartz was not recruited in 1997 and did not participate in a world war. Therefore, there were some other reasons for him to neglect the Sobolev book  which contains the principally new applications of distrubution theory to computer science. Sobolev based his pioneering results in numerical integration on developing the theory of the Fourier transform of distributions which was created by Schwartz.
Prudent in the appraisals, exceptionally tactful, and modest in his ripe years, Sobolev always abstained from any bit of details of the history of distribution theory neither in private conversations nor in his numerous writings. The opinion that he decided worthy to be left to the future generations about this matter transpires in his concise comments on the origins of distribution theory in the book [8, Ch. 8] which was printed in 1974:
Therefore, we can agree only in part with the following statement by Schwartz [11,p. 236]:
Most mathematicians agree that Israel Gelfand could be ranked as the best arbiter in distribution theory. The multi-volume series Generalized Functions written by him and his students was started in the mid 1950s and remains one of the heights of the world mathematical literature, the encyclopedia of distribution theory. In the preface to the first edition of the first volume of this series, Gelfand wrote:17
This is an accurate and just statement. We may agree with it.
Pondering over the fates of Sobolev and Schwart, it is impossible to obviate the problem of polarization of the opinions about the mathematical discovery of these scholars. The hope is naive that this problem will ever received a simple and definitive answer that satisfies and convinces everyone. It suffices to consider the available experience that concerns other famous pairs of mathematicians whose fates and contributions raise the quandaries that sometimes lasted for centuries and resulted in the fierce clashes of opinions up to the present day. The sources of these phenomena seem of a rather universal provenance that is not concealed in the particular personalities but resides most probably in the nature of mathematical creativity.
Using quite a risky analogy, we may say that mathematics has some features associated with the trends of artistry which are referred to traditionally as classicism and romanticism. It is hard to fail discerning the classic lineaments of the Hellenistic tradition in the writings of Euclid, Newton, Bolyai, Hilbert, and Bourbaki. It is impossible to fail to respond to the accords of the romantic anthem of the human genius which sound on the pages of the writings of Diophant, Leibniz, Lobachevsky, Poincaré, and Arnold.
The magnificent examples of mathematical classicism and romanticism glare from the creative contributions of Sobolev and Schwartz. These giants and their achievements will remain with us for ever.
February 4, 2008
The author thanks V. A. Aleksandrov and V. P. Golubyatnikov who helped him in better understanding of French sources. The author is especially grateful to Yu. L. Ershov who was persistent in inviting the author to make a talk at the special session of the Academic Council of the Sobolev Institute on October 14, 2003. The present article bases on this talk. The author acknowledges the subtle and deep comments of V. I. Arnold and V. S. Vladimirov on the preprint of a draft of the talk which led to many improvements.
2It was A. M. Vershik and V. I. Arnold who attracted the author's attention to the especial role of Günter in the prehistory of distribution theory.
3This is a cliché with a century-old history. The famous Russian symbolist Alexander Blok (1880–1921) used the concept of random feature in his incomplete poem “Revenge”as of 1910 [7,p. 482]. The prologue of this poem contains the lines that are roughly rendered in English as follows:
You share the gift of prudent measure For what keen vision might perceive. Erasing random features, treasure The world of beauty to receive.
5Also known as “generalized functions of slow growth.”
6Some historical details are collected in . Also see . J.-M. Kantor kindly made his article available to the author before publication with a courteous cooperation of Ch. Davis, Editor-in-Chief of The Mathematical Intelligencer. It was the proposal of Ch. Davis that the article by J.-M. Kantor be supplemented with the short comments  and .
11These are references to the articles in Sbornik [2, 3].
12Ibid., p. 1077.
14Published in 1950 by Leningrad State University, reprinted in 1962 by the Siberian Division of the Academy of Sciences of the USSR in Novosibirsk, and translated into English by the American Mathematical Society in 1963. The third Russian edition was printed by the Nauka Publishers in 1988.
16Cp. [25, p. 355]. This is a curious misprint: the correct reference to Schwartz's two-volume set should be .
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