# Walter Rudin

### Quick Info

Vienna, Austria

Madison, Wisconsin, USA

**Walter Rudin**was an Austrian-born American mathematician who worked in complex and harmonic analysis and is also known for his mathematical analysis textbooks.

### Biography

**Walter Rudin**'s great grandfather was Aron Pollak who made his money manufacturing matches. He was knighted in 1869 and chose the title von Rudin. Aron Pollak von Rudin's son, Alfred, took over the match business. He married Sara Lise Levi and the third of their children, Robert (born 7 January 1891) was Walter's father. Robert was an engineer and inventor, interested principally in sound recording and radio technology. He received a doctorate in engineering from the Technical University of Vienna in June 1914, just before the outbreak of World War I. Robert Pollak von Rudin served in the war then, on 1 July 1920, he married Natalie Adlersberg (known as Natasza). She was the daughter of Felix Adlersberg and Charlotte Aszkenasy. Walter was born in Vienna, the eldest of his parents' two children, having a younger sister Vera born 31 May 1925. The family was Jewish, but they were not religious people.

Walter did not attend kindergarten but, at age six, entered the Volksschule in Vienna. He spent four years at the Volksschule and then entered the same Realschule in Vienna that his father and grandfather had attended. The school [1]:-

... was within easy walking distance from our apartment. I had to take an entrance exam, which was no problem. We had 5 classes every day, from 8 to 1, with a 20 minute recess in the middle of the morning, when we could run around the gravel yard or play some games there. School met 6 days a week. All students and teachers were male. I think that the only females who ever entered the building were mothers on go-to -school nights. There were about 20 or 25 boys in my class, and we stayed together from one year to the next. We stayed in one room, and the teachers took turns coming in. Many of them had Ph.D's; all were called professor. The curriculum was set. There was no choice, except that electives could be added to what was required.The 1930s were difficult times for those of Jewish descent in Vienna with anti-Semitic attacks becoming more frequent. The policies of the Nazis in Germany after Hitler came to power in 1933 were a cause of grave concern since there were many Nazi sympathisers in Vienna. These worries, however, reached a quite different level when German troops marched into Austria and annexed the country on 12 March 1938. By the following day Jews were being harassed in Vienna, their homes and shops being attacked. Immediately, Walter's parents sent Walter and his sister Vera to Switzerland to continue their schooling. They arrived in Zürich and from there Vera went to Chexbres where she attended a girls' boarding school while Walter went to the Institut auf dem Rosenberg for boys in St Gallen [1]:-

At the Institute I was enrolled in a special small program which prepared its students for an exam administered by Oxford University and was given in many places all over the world.After about six months, Walter's parents managed to get out of Austria and reach Switzerland. They had to leave without taking any possessions other than the clothes they wore. On reaching Switzerland they hoped that this would become their permanent home but it was not long before they realised that this was not going to be possible. Walter's father had a friend in Paris, an engineer named M Givelet, and he helped arrange work for Robert as well as visas for the whole family. They moved to France where Robert began working for the Signal Corps but they had to frequently go to the Police to renew their permission to stay. They lived in Paris and in Paramé, Saint-Malo, but with the outbreak of World War II in 1939, the French were unhappy to have Germans in their country and Walter and his father Robert were sent to the Meslay-du-Maine internment camp. After being moved camps, Walter was given call-up papers for the French Army and sent to Pontivy. However, by this time the Germans had invaded France and soon fighting stopped. The French told Walter that he should run off which he did and managed to reach Saint Jean-de-Luz in the south west of France. From there he got a boat to England where he joined the Pioneer Corps of the British Army on 20 November 1940. This was the only part of the British Army that was open to foreigners. Rudin spent the war years in England. Towards the end of the war the Royal Navy announced that it needed interpreters, preferably native German speakers. After applying, he underwent tests in London and, on 4 February 1944 joined the Royal Navy.

In January 1945 he was given permission to travel to Avignon were his parents and sister were living. He returned to England, but the rest of his family were able to emigrate to the United States after the war ended. His sister Vera moved to Durham, North Carolina, where she undertook work on chemistry at Duke University. Walter Rubin was able to join the rest of his family in the United States towards the end of 1945 after the war ended. He joined his sister in Durham, North Carolina [2]:-

... and while he was there he talked to the people in the mathematics department. He had never gone to college, but he persuaded them that maybe he could be a junior.Rudin was awarded a B.A. by Duke University in 1947 and undertook research for his doctorate advised by John Jay Gergen, who had been a student of Griffith Conrad Evans and Szolem Mandelbrojt. Rudin was awarded his doctorate in 1949 for his thesis

*Uniqueness Theory for Laplace Series.*Rudin attached the following note:-

Submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences of Duke University. The author wishes to acknowledge his indebtedness to Professor J J Gergen, under whose direction the thesis was written.Rudin published two papers directly based on his thesis. The first part of the thesis was published as

*Integral representation of continuous functions*(1950). Here is Rudin's own summary of the results:-

It was shown by F Riesz in 'Sur les fonctions subharmoniques et leur rapport à la théorie du potentiel' Ⓣ (1930) that every subharmonic function u can be represented as the sum of the potential of its mass distribution plus a harmonic function; the potential appears in the form of a Stieltjes integral. We prove that the Stieltjes integral may be replaced by a Lebesgue integral if u is continuous, and if the lower generalized Laplacian of u is less than +∞, except possibly on a set of capacity zero. In other words, the above assumptions imply the absolute continuity of the mass distribution associated with u.Other results from the thesis appeared in a paper with the same title as his thesis.

Gergen offered him an Instructorship for 1949-50 at Duke University teaching 12 hours per week which he accepted. During this year at Duke University, Rudin met Mary Ellen Estill who had received a Ph.D. from the University of Texas in 1949 and was then appointed as an Instructor in Mathematics at Duke University. Mary Ellen wrote [2]:-

So we were both fresh young Ph.D.'s. We went together ... I think there were times during that year when I was interested in marrying Walter and there were times when he was interested in marrying me, but it was never at the same time. We were both just starting out in life.Rudin attended the International Congress of Mathematicians held in Cambridge, Massachusetts 30 August to 6 September 1950. He presented the paper

*Uniqueness theory for Hermite series*to the Congress on 1 September. He left Duke University in 1950 when he was appointed to a C.L.E. Moore Instructorship at the Massachusetts Institute of Technology. While at M.I.T. he completed the manuscript of his first book

*Principles of Mathematical Analysis*in the spring of 1952. The book was published in 1953 and reviewed by U S Haslam-Jones who wrote:-

The author deals with the classical elements of real variable theory: Dedekind real numbers, elementary set-theory, convergence, continuity, differentiation, the Riemann-Stieltjes integral, uniform convergence, functions of several variables, Lebesgue measure and integrals. The presentation is intended for the undergraduate (or immediately post-graduate) reader, and is clear and concise: the necessity for restrictive hypotheses in theorems is emphasised by illustrative counter-examples. In general, the spaces considered are Euclidean, but the reader is introduced to more abstract ideas: for example, general metric spaces and measure spaces.Kenneth May, reviewing the same book, writes [10]:-

This well-written book is an up-to-date, concise treatment of the foundations of classical real analysis (real numbers, set theory, limit processes, etc.) and two "modern" topics, the Stone-Weierstrass theorem and Lebesgue integration. At the advanced undergraduate or graduate level for which it is intended, it is self-contained. It will serve as a good text for courses at this level and, for those prepared to "dig," it would be a good choice for independent study of the fundamental ideas underlying calculus.Rudin was applying for positions in the spring of 1952 which was not easy since academic jobs were scarce. However, he was interviewed by the University of Maryland and the University of Rochester and appointed as an assistant professor at Rochester taking up the appointment in 1952. Rudin had met Mary Ellen Estill on various occasions after leaving Duke University and they decided to get married, which they did in August 1953 in Houston (where Mary Ellen was living) [1]:-

Vera was the bridesmaid and Joe, Ellen's brother, was best man. We spent a 4 or 5 day honeymoon in Galveston, then came back to Houston, squeezed all our wedding presents into my car, and started on the long trek to Rochester.Mary Ellen and Walter Rudin had four children: Catherine, Eleanor, Robert and Charles.

In 1958 a Sloan Fellowship enabled him to spend time at Yale University. While there he was contacted by R H Bing from the University of Wisconsin-Madison who offered him a summer teaching position. Rudin was not interested in the summer position but writes [1]:-

I was taken by surprise, said something like "I don't really want to teach summer school, I have a Sloan Fellowship, I don't need the money, ... " and then my brain slipped out of gear but my tongue kept on talking and I heard it say 'but how about a real job?'. My subconscious obviously didn't want to go back to Rochester. A couple of weeks later I was in Madison, being interviewed by Dean Mark Ingraham, a mathematician, and by President Conrad A Elvehjem.Rudin was appointed as a professor at the University of Wisconsin-Madison in 1959. He was later honoured by being appointed to a Vilas Professorship, the University's most prestigious professorship for scholarly achievement and the advancement of learning. He remained there until he retired in 1991 being made Vilas Professor Emeritus at that time.

As we have seen, Rudin's early work was on trigonometric series and holomorphic functions of one complex variable. However, by the late 1950s he was working on harmonic analysis on locally compact Abelian groups. In 1962 he published

*Fourier Analysis on Groups*. Stephen E Puckette writes in a review [14]:-

All in all, the author has assembled a first-rate treatise. It is one of the finest examples of classical analysis in the sense that abstract methods have been used effectively for an intense investigation of important concrete algebras.Jean-Pierre Kahane writes [6]:-

The author is known to be an excellent expositor, and he proves it once more by providing a considerable amount of information in less than three hundred pages without giving anywhere the impression of hurrying or pressing the reader.Edwin Hewitt explains that the book:-

... contains a huge amount of elegant analysis, presented in a brief compass. Possibly it is not a book for the beginner, but it is required reading for every harmonic analyst.Let us look briefly at the other books which Rudin published. There was the graduate text

*Real and complex analysis*(1966) which was reviewed by Robert Edmund Edwards:-

The more generally significant of the two aims of this book for first year graduate students is "to do away with the outmoded and misleading idea that analysis consists of two distinct halves, 'real variables' and 'complex variables'". While the author is not alone in adopting this aim, he here pursues it more vigorously and in greater detail than the reviewer has found elsewhere. In the reviewer's view, he goes a very long way toward fulfilment in a way which will carry many readers along with him.Writing about the same book, Victor Shapiro noted [16]:-

This book excels primarily in two important respects. The first is that the choice of topics serves as a superior introduction into much of what is current in analysis, in particular to the branches of harmonic analysis, partial differential equations, several complex variables, and Banach algebras. The second is that it blends both the concrete and abstract viewpoints and tends to do away with the notion that prevailed in the past separating analysis into "soft" analysis versus "hard" analysis.In the late 1960s, Rudin's interests turned towards questions in the theory of several complex variables. This was reflected in his next book

*Function theory in polydiscs*(1969) which, like all Rudin's books, received a very positive review:-

The theory of holomorphic functions on the unit disk has been studied extensively and has produced a wealth of famous theorems. Only recently, after a long period of neglect, the analogous questions for holomorphic functions on the unit polydisk have found interest. This book gives an excellent, up-to-date and easily readable account of these results in several variables, many of which are due to the author. The book is concerned with boundary values, factorization, distribution of zeros, invariant subspaces, interpolations and embeddings. ... The book provides an easy access to the subject matter and is indispensable for the investigator in this field.At the NSF regional conference from 1-5 June 1970, held at the University of Missouri, St Louis, Rudin gave a series of lecture which were published as

*Lectures on the edge-of-the-wedge theorem*(1971). His next book

*Functional analysis*(1973) again showed his skills as an expositor. Frank Smithies writes:-

This book is directed primarily towards the analytic rather than the algebraic or topological aspects of functional analysis. It is in some respects a sequel to the author's earlier book [Real and complex analysis], with whose more advanced parts there is some overlap. It is written in a very readable and not over-formal style; the author has avoided the temptation to organize his material so logically that it becomes thoroughly boring. Its contents form an excellent basis for more advanced work in topics on the analytic side of functional analysis.Reviewing the same book, Richard Kadison writes [6]:-

The book is excellent. Written by an expert - contributor to as well as expositor of subjects in analysis - it develops important topics and uses them for splendid applications.Rudin's final two books were

*Function theory in the unit ball of*$\mathbb{C}^{n}$ (1980), and

*New constructions of functions holomorphic in the unit ball of*$\mathbb{C}^{n}$ (1986). His skill as a writer led to him receiving the Leroy P Steele Prize for Mathematical Exposition from the American Mathematical Society in 1993. He was also honoured by the country of his birth with the award of an honorary degree from the University of Vienna in 2006.

### References (show)

- W Rudin,
*The Way I Remember It*(*American Mathematical Society, Providence, RI; London Mathematical Society, London,*1997). - D J Albers, C Reid and M E Rudin, An Interview with Mary Ellen Rudin,
*College Mathematics Journal***19**(2) (1988), 114-137. - J Cufi, On the death of Walter Rudin (Catalan),
*SCM Not. No.***29**(2010), 8-9. - G A Garreau, Review: Real and complex analysis, by Walter Rudin, J. Roy. Statist. Soc. Series D 36 (4) (1987), 423.
- R P Gillespie, Principles of Mathematical Analysis by Walter Rudin,
*The Mathematical Gazette***39**(329) (1955), 258-259. - R V Kadison, Review: Functional analysis, by Walter Rudin,
*Amer. Scientist***61**(5) (1973), 604. - J-P Kahane, Review: Fourier analysis on groups, by Walter Rudin,
*Bull. Am. Math. Soc.***70**(2) (1964), 230-232. - S G Krantz, Review: Function theory in the unit ball of Cn. by Walter Rudin,
*Bull. Am. Math. Soc. N.S.***5**(3) (1981, 331-339. - Mathematical Analyst Walter Rudin Dies at 89,
*Mathematical Association of America, Mathematical Sciences Digital Library*. http://mathdl.maa.org/mathDL/?pa=mathNews&sa=view&newsId=866 - K O May, Review: Principles of Mathematical Analysis, by Walter Rudin,
*The Mathematics Teacher***46**(8) (1953), 610. - A Nagel and EL Stout, Remembering Walter Rudin,
*Notices Amer. Math. Soc.***60**(3) (2013), 295-301. - Obituary: Walter Rudin,
*University of Wisconsin, Department of Mathematics*. http://www.math.wisc.edu/oldhome/news/WRudinObit.html - D Pareja, Walter Rudin (1921-2010), Lect. Mat. (1) (2011), 43-45.
- S E Puckette, Review: Fourier Analysis on Groups, by Walter Rudin,
*Amer. Math. Monthly***72**(6) (1965), 686-687. - R A Rankin, Review: Real and complex analysis, by Walter Rudin,
*The Mathematical Gazette***52**(382) (1968), 412. - V L Shapiro, Review: Real and complex analysis, by Walter Rudin,
*Bull. Am. Math. Soc.***74**(1) (1968), 79-83. - E L Stout, Function Theory in Polydiscs, by Walter Rudin,
*Amer. Scientist***58**(6) (1970), 706. - D Ziff, Noted UW-Madison mathematician Rudin dies at 89,
*Wisconsin State Journal*(21 May, 2010).

### Additional Resources (show)

Other websites about Walter Rudin:

### Honours (show)

Honours awarded to Walter Rudin

Written by J J O'Connor and E F Robertson

Last Update May 2013

Last Update May 2013