# Paul Richard Halmos

### Quick Info

Born
3 March 1916
Budapest, Hungary
Died
2 October 2006
Los Gatos, California, USA

Summary
Paul Halmos was a Hungarian-American mathematician who made fundamental advances in the areas of logic, probability, statistics, operator theory and functional analysis. He is best known for some of his text-books and for his collection of mathematicians' photographs.

### Biography

Paul Halmos's parents were Sándor Halmos (born 19 January 1881) and Paula Rosenberg (1888-1916). Sándor and Paula were married in 1903 and they had three children, George (born about 1909), John (born about 1911) and Paul (the subject of this biography). Paul's mother, Paula, 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 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. In the United States, Sándor Halmos worked for a year as an intern in a hospital in Omaha before moving to Chicago where he set up his own practice.

After five years in the United States, Paul's father, known in America as Alexander Charles Halmos, became a naturalised American citizen and, at that time, brought Paul from Hungary to join him in Chicago. The information that Paul came to the United States in 1929 after his father became an American citizen comes from his own account of his life but the 1930 Census states that Paul Halmos emigrated to the United States in 1924. This must be an error in filling in the Census form. Somewhat more puzzling is the fact that the 1930 Census form gives N/A (not available) for the year in which Paul's two older brothers entered the United States. According to Paul they came to join their father before he did. Once Alexander Halmos had established himself in Chicago he had remarried. His second wife, Irene, was a widow with two daughters Alice Reich (born in Illinois around 1917) and Jean Reich (born in Illinois around 1920). Irene had been born in Hungary in about 1897 and had first married at the age of 16. At the time of the 1930 Census, the family was living in Fullerton Avenue, Chicago, Cook, Illinois in a house that Alexander Halmos owned. They had two servants, Paul Tschampel, born in Germany around 1898, and his wife Tillie Tschampel, born around 1902 in what later became Czechoslovakia.

Paul Halmos attended school in Budapest up to the age of thirteen. He said [3]:-
In mathematics classes, I usually was above average. I was bored when class was going on, and I did things like take logarithms of very large numbers for fun.
After reaching the United States, 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 [3]):-
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.
He did remarkably well to cope with the move since when he arrived in Chicago he spoke Hungarian and German but no English. On his first day he managed to work out where to go by exchanging a few words with a teacher in Latin and French. After six months he understood English well and could speak "rapid, incorrect, ungrammatical, colloquial English."

While still fifteen years old he entered the University of Illinois to study chemical engineering. He had considered other options such as studying law at a law school but opted for chemistry. His age was not a problem, he said ([1] or [3]):-
I was tall for my age and cocky. I pretended to be older and got along fine.
After one year he became disappointed with chemistry, saying he got his hands dirty, so he changed to mathematics and philosophy but did not particularly shine at mathematics ([1] or [3]):-
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, and 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. This came about mainly because he had preformed poorly in the oral comprehensive examination for the Masters' Degree in philosophy. It was in September 1935 that he taught his first course, namely freshman algebra [36]:-
... its purpose was to reveal the secrets of quadratic equations (for which there was a formula) and parentheses (which were abominable entities and had to be eliminated at the drop of a hat). The course met at 8:00 in the morning, five days a week - yes, five days, Monday through Friday, inclusive; my pay was $45.00 a month. Incidentally, I was living at the time in an old-fashioned, comfortable, large, 5-room apartment, within five minutes walk of the campus; the rent was$45.00 a month.
After thinking that algebra was the right mathematical topic for him, he quickly changed to analysis and studied for his Ph.D. under Joseph Leo Doob. This was awarded in 1938 for his thesis on measure-theoretic probability Invariants of Certain Stochastic Transformation: The Mathematical Theory of Gambling Systems. This topic led to him becoming involved in a debate about gambling which was going on in Urbana-Champaign at the time and, after he was interviewed by a local newspaper, they ran the headline "You Can't Win, Says P R Halmos."

Jobs were not so easy to come by ([1] or [3]):-
I typed 120 letters 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 in 1938-39 I had a job, but I kept applying.
In February 1939 Halmos was successful in obtaining a post at Reed College in Oregon. He accepted the position but in April his friend Warren Ambrose was offered a scholarship at the Institute for Advanced Study in Princeton. Halmos wrote ([1] or [3]):-
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 [2]:-
... his speed, plus depth, plus insight, plus inspiration turned me on.
They wrote one joint paper while Halmos was his assistant. Halmos explains in [35]:-
After the thinking and the talking were finished, it became my job to do the writing. I did it, and I submitted to him a typescript of about 12 pages. He read it, criticized it mercilessly, crossed out half, and rewrote the rest; the result was about 18 pages. I removed some of the Germanisms, changed a few spellings, and compressed it into 16 pages. He was far from satisfied, and made basic changes again; the result was 20 pages. The almost divergent process continued (four innings on each side as I now recall it); the final outcome was about 30 typescript pages (which came to 19 in print).
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. For extracts from reviews of this book and several others of Halmos's books see THIS LINK.

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. In 1945 he married Virginia Templeton Pritchett. Virginia had been born on 21 December 1915 in Omaha, Nebraska and had studied at Vassar College followed by graduate study in logic and the foundations of mathematics at Brown University.

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 the summer of 1955 the University of Chicago held a functional analysis meeting with George Mackey, Irving Segal, Irving Kaplansky and Paul Halmos as lecturers. Lawrence Wallen was in the audience and describes Halmos's lectures in [60]:-
Many words and phrases were clipped, others almost drawled and the timbre was pleasantly resonant. In fact, the total effect was rather musical. The vestigial Hungarian accent Paul bemoans was so slight as to seem just a mannerism ... The lectures proceeded at a no-nonsense but unhurried pace.
Late in the summer of 1955, Wallen [60]:-
... met Virginia (Ginger) Halmos, a striking woman I recall thinking. Any picture of Paul that omits Ginger is grossly incomplete. In the first place, she's crucial to keeping the entropy of the Halmos household improbably small and in keeping Paul and the cats hale and hearty. This, of course, doesn't define Ginger. She's a woman of remarkable intelligence with a fine wit that not everyone is privy to. She's the ecological Halmos who fishes floundering lizards from the pool and worries about wetlands. She still can be seen riding her bike in the perilous environs of San Jose and not infrequently sports a Band Aid from a minor contretemps. Paul frequently, and not without cause, frets about her safety.
In 1961 Halmos moved to the University of Michigan [60]:-
Michigan seemed an idyllic situation for Paul. There were good friends, good colleagues, good students. There was an active social life, a fine home, and a terrific Old English sheep dog (Bertrand Russell by name). There was good walking and there was even that miserable climate (damp cold winters and damp hot summers) on which he seemed to thrive best.
In 1965 Halmos was one of three plenary speakers at the British Mathematical Colloquium in Dundee, Scotland (the other two were Claude Chevalley and Arthur Erdélyi). Halmos gave the lecture Some recent progress in Hilbert space. He was introduced by Frank Bonsall as follows (see [2]):-
Professor Halmos may look like one mathematician, but in reality be is an equivalence class and has worked in several fields including algebraic logic and ergodic theory; this afternoon his representative from Hilbert space will speak to us.
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 [26]:-
When he was at Indiana, he organized a seminar for mathematicians from Urbana, Lafayette, Bloomington, and all points in between. It continues to this day as the 'Wabash Extramural Functional Analysis Seminar'.
He remained at Indiana until 1985 when he moved to Santa Clara. G L Alexanderson writes in [1]:-
In early 1984 I 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, Algebraic logic (1962), A Hilbert space problem book (1967) and Lectures on Boolean algebras (1974). For some extracts from reviews of some of Halmos's books see THIS LINK, THIS LINK and THIS LINK.

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. In fact he received the Chauvenet prize, the Polya prize, and two Lester R Ford awards from the Mathematical Association of America. For extracts from some of Halmos's "popular" papers on writing mathematics, teaching, the nature of mathematics etc, see THIS LINK.

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 was a frequent visitor to Scotland. He attended regularly the four-yearly St Andrews Colloquium. We 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 was elected a Fellow of the Royal Society of Edinburgh. He has also been awarded an honorary D.Sc. from the University of St Andrews.

Together with his wife, Halmos contributed \$4,000,000 for the rebuilding of the Carriage House Conference Center in Washington D.C. in 2002. He also funded Mathematical Association of America programmes at the Carriage Center. He donated funds to support the Mathematical Association of America's Euler Prize and its Halmos-Ford Prize for expository writing. Paul and Virginia Halmos also donated funds to the American Mathematical Society to set up a J L Doob Prize for outstanding expository mathematical writing.

Halmos died in Los Gatos, California at the age of ninety after contracting pneumonia. His wife Virginia continued to live in Los Gatos where she died, aged 99, on 19 January 2015. Gerald L Alexanderson writes:-
In her last weeks she was still listening to drafts of an upcoming book on G H Hardy and reminiscing about J E Littlewood (whom, she was proud to say, she had been seated next to at high table at Trinity), Frank Smithies and others of Hardy's era.
Let us end this biography by quoting Halmos's reply when asked what mathematics meant to him:-
It is security. Certainty. Truth. Beauty. Insight. Structure. Architecture. I see mathematics, the part of human knowledge that I call mathematics, as one thing - one great, glorious thing.

### References (show)

1. J H Ewing and F W Gehring (eds.), Paul Halmos : Celebrating 50 years of mathematics (New York, 1991).
2. P Halmos, I Want to Be a Mathematician: An Automathography (Springer-Verlag, New York, 1985).
3. D J Albers, Paul Halmos: maverick mathologist, Two-Year College Math. J. 13 (4) (1982), 226-242.
4. D J Albers, In touch with God: an interview with Paul Halmos, College Math. J. 35 (1) (2004), 2-14.
5. S Axler, Paul Halmos and Toeplitz operators, in Paul Halmos (New York, 1991), 257-263.
6. J Baylis, Review: I Want to Be a Mathematician: An Automathography, by Paul R Halmos, The Mathematical Gazette 70 (453) (1986), 253-255.
7. Bibliography of Paul Halmos, in Paul Halmos (New York, 1991), 61-69.
8. A Borgers, Review: Naive set theory, by Paul R Halmos, The Journal of Symbolic Logic 34 (2) (1969), 308.
9. J R Buchi, Review: Naive set theory, by Paul R Halmos, Philosophy of Science 28 (4) (1961), 445.
10. S D Comer, Review: Logic as algebra, by Paul Halmos and Steven Givant, The Journal of Symbolic Logic 63 (4) (1998), 1604.
11. J B Conway, Paul Halmos and the progress of operator theory, in Paul Halmos (New York, 1991), 155-167.
12. J L B Cooper, Review: Finite dimensional vector spaces (2nd edition), by Paul R Halmos, The Mathematical Gazette 44 (348) (1960), 142-143.
13. J L B Cooper, Review: Measure theory, by Paul R Halmos, The Mathematical Gazette 35 (312) (1951), 142.
14. J L B Cooper, Review: Introduction to Hilbert space and theory of spectral multiplicity, by Paul R Halmos, The Mathematical Gazette 36 (317) (1952), 218-219.
15. A Dijksma, Paul R Halmos : a complete professional mathematician, Nieuw Arch. Wisk. (4) 13 (1) (1995), 49-60.
16. N Dobrinen, Review: Logic as algebra, by Paul Halmos and Steven Givant, The Bulletin of Symbolic Logic 16 (2) (2010), 281-282.
17. J A Dossey, Review: I Want to Be a Mathematician: An Automathography, by Paul R Halmos, The Mathematics Teacher 79 (6) (1986), 481-482.
18. Y N Dowker, Review: Lectures on ergodic theory, by Paul R Halmos, Bull. Amer. Math. Soc. 65 (4) (1959), 253-254.
19. J Ewing, Paul Halmos: no apologies, in A century of advancing mathematics (Math. Assoc. America, Washington, DC, 2015), 411-414.
20. J Ewing, John Paul Halmos: in his own words, in A glimpse at Hilbert space operators (Birkhäuser Verlag, Basel, 2010), 11-25.
21. J Ewing, John Paul Halmos: in his own words, Notices Amer. Math. Soc. 54 (9) (2007), 1136-1144.
22. P Fillmore and N Higson, Review: A Hilbert space problem book (2nd edition), by Paul R Halmos, Amer. Math. Monthly 91 (9) (1984), 592-594.
23. L Garrison, Review: Problems for Mathematicians Young and Old, by Paul R Halmos, The Mathematics Teacher 85 (7) (1992), 592.
24. H M Gehman, Review: Naive set theory, by Paul R Halmos, Philosophy and Phenomenological Research 22 (1) (1961), 122-123.
25. H M Gehman, Review: Measure theory, by Paul R Halmos, Mathematics Magazine 26 (3) (1953), 173-174.
26. A M Gleason, Yueh-Gin Gung and Dr Charles Y Hu Award for Distinguished Service to Paul R Halmos, Amer. Math. Monthly 107 (3) (2000), 193-194.
27. R L Goodstein, Review: Naive set theory, by Paul R Halmos, The Mathematical Gazette 45 (354) (1961), 375.
28. F Q Gouvea, Review: I Want to Be a Mathematician: An Automathography, by Paul R Halmos, American Mathematical Association (2006). http://www.maa.org/node/105650
29. P R Halmos, How to write mathematics, Enseign. Math. (2) 16 (1970),
30. P R Halmos, How to talk mathematics, Notices Amer. Math. Soc. 21 (1974), 155-158.
31. P R Halmos, Four panel talks on publishing, Amer. Math. Monthly 82 (1975), 14-17.
32. P R Halmos, The problem of learning to teach, Amer. Math. Monthly 82 (1975), 466-476.
33. P R Halmos, The heart of mathematics, Amer. Math. Monthly 87 (1980), 519-524.
34. P R Halmos, Mathematics as a creative art, American Scientist 56 (1968), 375-389.
35. P R Halmos, The Legend of John Von Neumann, Amer. Math. Monthly 80 (4) (1973), 382-394.
36. P R Halmos, What is Teaching?, Amer. Math. Monthly 101 (9) (1994), 848-854.
37. L Henkin, Review: Algebraic logic, by Paul R Halmos, Science, New Series 138 (3543) (1962), 886-887.
38. A Heyting, Review: Naive set theory, by Paul R Halmos, Synthese 13 (1) (1961), 86-87.
39. G Hoare, Review: Logic as algebra, by Paul Halmos and Steven Givant, The Mathematical Gazette 84 (499) (2000), 172-173.
40. M Kac, Review: Finite dimensional vector spaces, by Paul R Halmos, Bull. Amer. Math. Soc. 49 (5) (1943), 349-350.
41. M R Lipman, Review: Logic as algebra, by Paul Halmos and Steven Givant, The Mathematics Teacher 92 (4) (1999), 371.
42. E R Lorch, Review: Introduction to Hilbert space and theory of spectral multiplicity, by Paul R Halmos, Bull. Amer. Math. Soc. 58 (3) (1952), 412-415.
43. N Lord, Review: Linear Algebra Problem Book (1995), by Paul R Halmos, The Mathematical Gazette 81 (490) (1997), 168-170.
44. P Maher, Review: A Hilbert space problem book (2nd edition), by Paul R Halmos, The Mathematical Gazette 73 (465) (1989), 259-260.
45. E Mendelson, Review: Naive set theory, by Paul R Halmos, The Journal of Philosophy 57 (15) (1960), 512-513.
46. R Messer, Review: Linear Algebra Problem Book (1995), by Paul R Halmos, Amer. Math. Monthly 105 (6) (1998), 577-579.
47. H Mirkil, Review: Naive set theory, by Paul R Halmos, Amer. Math. Monthly 68 (4) (1961), 392.
48. D Monk, Review: Algebraic logic, by Paul R Halmos, Amer. Math. Monthly 71 (6) (1964), 708-709.
49. J C Oxtoby, Review: Measure theory, by Paul R Halmos, Bull. Amer. Math. Soc. 59 (1) (1953), 89-91.
50. D Pareja Heredia, Obituary: Paul Richard Halmos (1916-2006) (Spanish), Lect. Mat. 27 (2) (2006), 163-166.
51. R S Pierce, Review: Lectures on Boolean algebras, by Paul R Halmos, The Journal of Symbolic Logic 31 (2) (1966), 253-254.
52. Publications of Paul R Halmos, in A glimpse at Hilbert space operators (Birkhäuser Verlag, Basel, 2010), 33-40.
53. H Radjavi and P Rosenthal, Obituary: Paul Halmos, 1916-2006, in A glimpse at Hilbert space operators (Birkhäuser Verlag, Basel, 2010), 27-29.
54. H Radjavi and P Rosenthal, Obituary: Paul Halmos, 1916-2006, CMS Notes 39 (2) (2007).
55. G-C Rota, Review: I Want to Be a Mathematician: An Automathography, by Paul R Halmos, Amer. Math. Monthly 94 (7) (1987), 700-702.
56. V S Sunder, Paul Halmos - expositor par excellence, in A glimpse at Hilbert space operators (Birkhäuser Verlag, Basel, 2010), 3-10.
57. V S Sunder, Paul Halmos - expositor par excellence, Ganita Bharati 28 (1-2) (2006), 193-199.
58. S Wagon, Review: Problems for Mathematicians Young and Old, by Paul R Halmos, Amer. Math. Monthly 99 (9) (1992), 888-890.
59. A D Wallace, Review: Lectures on Boolean algebras, by Paul R Halmos, Science, New Series 144 (3618) (1964), 531-532.
60. L J Wallen, Walking and Talking with Halmos, in Paul Halmos : Celebrating 50 years of mathematics (New York, 1991), 133-142.
61. A Wilansky, Review: Finite dimensional vector spaces (2nd edition), by Paul R Halmos, Amer. Math. Monthly 66 (6) (1959), 528-529.
62. S G Winter, Jr, Review: Naive set theory, by Paul R Halmos, Journal of the American Statistical Association 56 (296) (1961), 1022-1023.
63. A Zaanen, Review: Bounded integral operators on L2 spaces, by Paul Richard Halmos and Viakalathur Shankar Sunder, Bull. Amer. Math. Soc. (N.S.) 1 (6) (1979), 953-960.