**Paul Halmos**'s mother died when Paul was six months old. Paul's father was a successful physician in Budapest who had the rather remarkable foresight to realise the problems that were going to befall Europe. So in 1924 Paul's father emigrated to Chicago in the United States, leaving Paul and his two elder brothers in Budapest. There they were looked after by the physician who took over his father's practice.

After five years in Chicago, Paul's father became an American citizen and, at that time, brought Paul from Hungary to join him in Chicago. He attended high school in Chicago but rather remarkably he missed out four years schooling in the process. Halmos says that there was some confusion since in Hungary four years of primary schooling were followed by eight years of secondary schooling. He had completed seven of these twelve years but Halmos said ([1] or [2]):-

I hinted to the school authorities that I had completed three years of secondary school, and I was believed. ... a year and a half later, at the age of fifteen, I graduated from high school.

While still 15 he entered the University of Illinois to study chemical engineering. His age was not a problem, he said ([1] or [2]):-

I was tall for my age and cocky. I pretended to be older and got along fine.

After one year he changed to mathematics and philosophy but did not particularly shine at mathematics ([1] or [2]):-

I was a routine calculus student - I think I got B's. I did not understand about limits. I doubt that they taught it. ... But I was good at integrating and differentiating things in a mechanical sense. Somehow I like it. I kept fooling around with it.

Despite being so young when he entered his undergraduate course and despite changing from chemical engineering to mathematics and philosophy he still completed the four year degree in three years graduating in 1934. He began graduate studies at the University of Illinois at Urbana-Champaign, still with philosophy as his main subject, mathematics as his minor subject.

It was not until the end of the academic year 1935-36 that Halmos made the move from philosophy to mathematics. After thinking that algebra was the right subject for him, he quickly changed to analysis and studied for his Ph.D. under Doob. This was awarded in 1938 for his thesis on measure-theoretic probability *Invariants of Certain Stochastic Transformation: The Mathematical Theory of Gambling Systems*. Jobs were not so easy to come by ([1] or [2]):-

I typed120letters of application, mailed them out, and got two answers, both no. The University of Illinois took pity on me and kept me on as an instructor. So in38-39I had a job, but I kept applying.

In February 1939 Halmos was successful in obtaining a post at Reed College in Oregon. He accepted but in April his friend Warren Ambrose was offered a scholarship at the Institute for Advanced Study in Princeton. Halmos wrote ([1] or [2]):-

That made me mad. I wanted to go, too! I resigned my job, making the department head, whom I had never met, very unhappy, of course. I ... went to my father and asked to borrow a thousand dollars ... I wrote to Veblen and asked if I could become a member of the Institute for Advanced Study even though I had no fellowship. ... I moved to Princeton.

After six months Halmos was offered a fellowship, and in his second year at Princeton he became von Neumann's assistant. Ambrose writes in [1]:-

This was wonderful for Paul because he ... idolised von Neumann ... This seemed to have been the first time in Paul's career when he received what he deserved and I think it must have been one of the happiest times in his life.

Halmos said of von Neumann:-

... his speed, plus depth, plus insight, plus inspiration turned me on.

A debt that Halmos owes to von Neumann is that one of his lecture courses inspired Halmos's first book. In 1942 Halmos published *Finite Dimensional Vector Spaces* which was to bring him instant fame as an outstanding writer of mathematics.

After leaving the Institute for Advanced Study, Halmos was appointed to Syracuse University, New York. While in Syracuse he took part in teaching soldiers in the Army's Specialized Training Program. At the end of World War II Halmos decided it was time for a change and, in 1946, he became an assistant professor at the University of Chicago.

In 1961 Halmos moved to the University of Michigan. In 1968-69 he served for one year as chairman of the mathematics department of the University of Hawaii. At the end of the year he accepted a professorship at Indiana University. He remained at Indiana until 1985 when he moved to Santa Clara. G L Alexanderson writes in [1]:-

In early1984I received a telephone call from Paul Halmos ... during which he said, among other things, that he would like to be someplace with more sunny days. The Bloomington winter seemed long. ... I responded that ... I would think about it. When I had, I called him and asked him whether he might consider Santa Clara... I raised the question of his joining us at Santa Clara with some hesitation because, though we may have good weather, Santa Clara is not the kind of institution at which Paul had spent his career.

Halmos is known for both his outstanding contributions to operator theory, ergodic theory, functional analysis, in particular Hilbert spaces, and for his series of exceptionally well written textbooks. These include *Finite dimensional vector spaces* (1942), *Measure theory* (1950), *Introduction to Hilbert space and theory of spectral multiplicity* (1951), *Lectures on ergodic theory* (1956), *Entropy in ergodic theory* (1959), *Naive set theory* (1960), *Algebraic logic* (1962), *A Hilbert space problem book* (1967) and *Lectures on Boolean algebras* (1974).

In 1983 he received the Steele Prize for exposition from the American Mathematical Society. The citation read:-

The award for a book or substantial survey or research-expository paper is made to Paul R Halmos for his many graduate texts in mathematics, dealing with finite dimensional vector spaces, measure theory, ergodic theory and Hilbert space. Many of these books were the first systematic presentations of their subjects in English. Their felicitous style and content has had a vast influence on the teaching of mathematics in North America. His articles on how to write, talk and publish mathematics have helped all mathematicians to communicate their ideas and results more effectively.

Halmos has received many other awards for his writing and teaching. For example, in 1993, he received a Distinguished Teacher award from the Mathematical Association of America.

J B Conway writes in [1] about Halmos's contributions to operator theory:-

... Paul has a number of papers and theorems that anyone would be proud to call his own. But the thing that has always struck me about his work is the extraordinary number of topics and problems that are dominant themes in the current research of today and that have their origin in his work. Over the years Paul has demonstrated an uncanny ability to extract crucial properties from a given mathematical entity and lay it open before his colleagues in such a manner that there is a universal inclination to look and explore further.

Halmos has been a frequent visitor to Scotland. He attends regularly the four-yearly St Andrews Colloquium. I [EFR] first met him at the 1972 St Andrews Colloquium and fully agree with Alastair Gillespie's comments in [1]:-

These Colloquia are just the sort of things that Halmos relishes in - a happy mixture of expository mathematics and recreation - a mathematical holiday, in fact.

Halmos spent part of his 1973 sabbatical leave in Edinburgh and has been elected a Fellow of the Royal Society of Edinburgh. He has also been awarded an honorary D.Sc. from the University of St Andrews.

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

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