保罗·哈尔莫斯Paul Halmos

保罗·哈尔莫斯的父母是Sándor 哈尔莫斯（生于1881年1月19日）和Paula Rosenberg（1888-1916）。Sándor和Paula于1903年结婚，他们有三个孩子，George（约生于1909年）、弗瑞兹·约翰（约生于1911年）和哈尔莫斯（本传记的主人公）。哈尔莫斯的母亲Paula在哈尔莫斯六个月大时去世。哈尔莫斯的父亲是布达佩斯一位成功的医生，他有着相当非凡的远见，意识到即将降临欧洲的问题。因此，在1924年，哈尔莫斯的父亲移民到美国，将哈尔莫斯和他的两个哥哥留在布达佩斯。在那里，他们由接管他父亲诊所的医生照顾。在美国，Sándor 哈尔莫斯在奥马哈的一家医院做了一年实习医生，然后搬到芝加哥，在那里开设了自己的诊所。

在美国五年后，哈尔莫斯的父亲，在美国被称为Alexander Charles Halmos，成为归化的美国公民，并在那时将哈尔莫斯从匈牙利带到芝加哥与他团聚。关于哈尔莫斯在父亲成为美国公民后于1929年来到美国的信息来自他自己的生活叙述，但1930年人口普查显示哈尔莫斯于1924年移民到美国。这一定是填写普查表时的错误。更令人费解的是，1930年普查表在哈尔莫斯的两个哥哥进入美国的年份上给出了N/A（不可用）。根据哈尔莫斯，他们在他之前来与父亲团聚。一旦Alexander Halmos在芝加哥站稳脚跟，他就再婚了。他的第二任妻子Irene是一个寡妇，有两个女儿Alice Reich（约1917年生于伊利诺伊州）和Jean Reich（约1920年生于伊利诺伊州）。Irene约1897年生于匈牙利，16岁时第一次结婚。在1930年人口普查时，这家人住在伊利诺伊州库克县芝加哥富勒顿大道的一所房子里，房子是Alexander Halmos拥有的。他们有两个仆人，哈尔莫斯 Tschampel，约1898年生于德国，以及他的妻子Tillie Tschampel，约1902年生于后来成为捷克斯洛伐克的地方。

哈尔莫斯在布达佩斯上学直到十三岁。他说[3]：-

在数学课上，我通常高于平均水平。上课进行时我感到无聊，我会做一些像取非常大数的对数这样的事情来消遣。

到达美国后，他在芝加哥上高中，但相当引人注目的是，他在这个过程中错过了四年的学校教育。哈尔莫斯说，这有些混乱，因为在匈牙利，四年小学教育之后是八年中学教育。他完成了这十二年中的七年，但哈尔莫斯说（[1]或[3]）：-

我向学校当局暗示我已经完成了三年中学教育，他们相信了。……一年半后，十五岁时，我高中毕业了。

他应对这次搬迁表现得相当出色，因为当他到达芝加哥时，他会说匈牙利语和德语，但不会英语。第一天，他通过用拉丁语和法语与一位老师交换了几句话，设法弄清了该去哪里。六个月后，他很好地理解了英语，并能说“快速、不正确、不合语法、口语化的英语”。

还在十五岁时，他进入伊利诺伊大学学习化学工程。他曾考虑过其他选择，比如在法学院学习法律，但选择了化学。他说，他的年龄不是问题（[1]或[3]）：-

就我的年龄来说，我个子高，而且很自负。我假装自己年纪更大，并且相处得很好。

一年后，他对化学感到失望，说他弄脏了手，于是他转到了数学和哲学，但在数学方面并不特别出色（[1]或[3]）：-

我是一个普通的微积分学生——我想我得了B。我不理解极限。我怀疑他们教过这个。……但我在机械意义上擅长积分和微分。不知怎么的，我喜欢它。我一直摆弄它。

尽管他进入本科课程时如此年轻，尽管从化学工程转到数学和哲学，他仍然在三年内完成了四年的学位，于1934年毕业。他开始在伊利诺伊大学厄巴纳-香槟分校攻读研究生，仍然以哲学为主修，数学为辅修。

直到1935-36学年末，哈尔莫斯才从哲学转向数学。这主要是因为他在哲学硕士学位的口试综合考试中表现不佳。正是在1935年9月，他教授了他的第一门课程，即新生代数[36]：-

……其目的是揭示二次方程（有一个公式）和括号（这是可憎的实体，必须随时消除）的秘密。课程在早上8点上课，每周五天——是的，五天，周一到周五，包括；我的工资是每月45.00美元。顺便说一句，我当时住在一套老式、舒适、宽敞的五居室公寓里，步行五分钟即可到达校园；租金是每月45.00美元。

在认为代数是适合他的正确数学主题后，他很快转向分析，并在杜布的指导下攻读博士学位。1938年，他因关于测度论概率的学位论文Invariants of Certain Stochastic Transformation: The Mathematical Theory of Gambling Systems而获得博士学位。这个主题使他卷入了当时在厄巴纳-香槟正在进行的一场关于赌博的辩论，在他接受当地报纸采访后，他们打出了标题“你赢不了，P R 哈尔莫斯说。”

工作并不那么容易找到（[1]或[3]）：-

我打了120封求职信，寄了出去，收到两封回复，都是“不”。伊利诺伊大学可怜我，让我留校做讲师。所以1938-39年我有了一份工作，但我一直在申请。

1939年2月，哈尔莫斯成功获得了俄勒冈州里德学院的一个职位。他接受了这个职位，但在4月，他的朋友Warren Ambrose获得了普林斯顿高等爱德华·斯图迪的奖学金。哈尔莫斯写道（[1]或[3]）：-

这让我很生气。我也想去！我辞去了工作，当然，这让从未见过面的系主任非常不高兴。我……去找父亲，请求借一千美元……我写信给奥斯瓦尔德·维布伦，询问即使我没有奖学金，能否成为高等斯图迪的成员。……我搬到了普林斯顿。

六个月后，哈尔莫斯获得了奖学金，在普林斯顿的第二年，他成为了冯·诺伊曼的助手。Ambrose在[1]中写道：-

这对哈尔莫斯来说太棒了，因为他……崇拜冯·诺伊曼……这似乎是哈尔莫斯职业生涯中第一次得到他应得的东西，我认为这一定是他一生中最快乐的时光之一。

哈尔莫斯谈到冯·诺伊曼时说[2]：-

……他的速度、深度、洞见和灵感都让我着迷。

在哈尔莫斯担任他的助手期间，他们合写了一篇论文。哈尔莫斯在[35]中解释道：-

思考和讨论结束后，写作就成了我的任务。我完成了写作，并向他提交了一份约12页的打字稿。他读了之后，毫不留情地批评，划掉了一半，重写了其余部分；结果大约有18页。我去掉了一些德语式的表达，改了几个拼写，压缩到16页。他远不满意，又做了根本性的修改；结果变成了20页。这种几乎反复无常的过程持续着（我现在回想起来，双方各改了四轮）；最终结果是大约30页打字稿（印刷出来是19页）。

哈尔莫斯欠冯·诺伊曼的一份情是，他的其中一门讲座课程激发了哈尔莫斯的第一本书。1942年，哈尔莫斯出版了Finite Dimensional Vector Spaces，这本书使他作为杰出的数学作家一举成名。关于这本书以及哈尔莫斯其他几本书的书评摘录，见THIS LINK。

离开高等斯图迪研究院后，哈尔莫斯被任命到纽约锡拉丘兹大学。在锡拉丘兹期间，他参与了陆军专业训练计划中士兵的教学工作。1945年，他与弗吉尼亚·坦普尔顿·普里切特结婚。弗吉尼亚于1915年12月21日出生在内布拉斯加州奥马哈，曾就读于瓦萨学院，随后在布朗大学攻读逻辑和数学基础的 graduate study。

第二次世界大战结束时，II Halmos决定是时候做出改变了，于是在1946年，他成为芝加哥大学的助理教授。1955年夏天，芝加哥大学举办了一次泛函分析会议，George Mackey、尔文·席格、厄文·卡普兰斯基和哈尔莫斯担任讲师。Lawrence Wallen在听众席中，他在[60]中描述了哈尔莫斯的讲座：-

许多单词和短语被省略了，另一些几乎拖长了声音，音色悦耳而富有共鸣。事实上，整体效果相当有音乐性。哈尔莫斯所哀叹的残留匈牙利口音如此轻微，以至于似乎只是一种习惯……讲座以严肃但不慌不忙的节奏进行。

1955年夏末，Wallen [60]：-

……遇见了Virginia（Ginger）哈尔莫斯，我记得当时觉得她是一位引人注目的女性。任何略去Ginger的哈尔莫斯画像都是极不完整的。首先，她对于使哈尔莫斯家的熵保持低得不可思议，以及使哈尔莫斯和猫们健壮活泼，都至关重要。当然，这并不能定义Ginger。她是一位才智出众的女性，有着并非人人都能领略的绝妙机智。她是生态方面的哈尔莫斯，会从泳池里捞起挣扎的蜥蜴，并担忧湿地。人们仍能看到她在圣何塞危险的环境中骑自行车，而且她常常因小磕碰而贴着创可贴。哈尔莫斯经常——并非没有理由地——为她的安全担忧。

1961年，哈尔莫斯搬到了密歇根大学 [60]：-

对哈尔莫斯来说，密歇根似乎是一个田园诗般的地方。那里有好友、好同事、好学生。有活跃的社交生活、一座好房子，还有一只极棒的古代英国牧羊犬（名叫伯特兰·罗素）。那里适合散步，甚至还有那种糟糕的气候（湿冷的冬天和湿热夏天），而他似乎在那样的气候中最为茁壮。

1965年，哈尔莫斯是苏格兰邓迪举行的英国数学学术讨论会的三位全会演讲者之一（另外两位是谢瓦莱和Arthur Erdélyi）。哈尔莫斯作了题为Some recent progress in Hilbert space.的演讲。他由Frank Bonsall介绍如下（见[2]）：-

哈尔莫斯可能看起来像一位数学家，但实际上他是一个等价类，并曾在包括代数逻辑和遍历理论在内的多个领域工作；今天下午，他来自希尔伯特空间的代表将向我们演讲。

1968-69年，他担任夏威夷大学数学系主任一年。年底，他接受了印第安纳大学的教授职位[26]：-

在印第安纳大学期间，他为来自厄巴纳、拉法叶、布卢明顿及其中间各地的数学家组织了一个研讨会。这个研讨会至今仍在继续，名为“沃巴什校外泛函分析研讨会”。

他留在印第安纳大学直到1985年，然后搬到圣克拉拉。G L Alexanderson在[1]中写道：-

1984年初，我接到哈尔莫斯的电话……在通话中，他提到，除其他事项外，他希望能去一个阳光更充足的地方。布卢明顿的冬天似乎很漫长。……我回答说……我会考虑一下。考虑之后，我打电话问他是否可以考虑圣克拉拉……我有些犹豫地提出让他加入我们在圣克拉拉，因为尽管我们可能天气好，但圣克拉拉不是哈尔莫斯度过职业生涯的那种机构。

哈尔莫斯以其在算子理论、遍历理论、泛函分析，特别是大卫·希尔伯特空间方面的杰出贡献，以及一系列写得异常出色的教科书而闻名。这些教科书包括Finite dimensional vector spaces（1942年）、Measure theory（1950年）、Introduction to Hilbert space and theory of spectral multiplicity（1951年）、Lectures on ergodic theory（1956年）、Entropy in ergodic theory（1959年）、Naive set theory、Algebraic logic （1962年）、 A Hilbert space problem book（1967年）和Lectures on Boolean algebras（1974年）。关于哈尔莫斯一些书籍评论的摘录，见THIS LINK和THIS LINK。

1983年，他因阐述工作获得了美国数学会颁发的凯瑟琳·斯蒂尔奖。颁奖词写道：-

授予哈尔莫斯 R 哈尔莫斯 图书或重要综述或研究阐述论文奖，以表彰他在数学研究生教材方面的众多著作，这些著作涉及有限维向量空间、测度论、遍历论和希尔伯特空间。其中许多书是这些主题首次以英语进行的系统阐述。其恰当的风格和内容对北美的数学教学产生了巨大影响。他关于如何写作、演讲和发表数学文章的文章帮助所有数学家更有效地交流他们的思想和成果。

哈尔莫斯 因其写作和教学还获得过许多其他奖项。例如，1993年，他获得了美国数学协会颁发的杰出教师奖。事实上，他获得了肖维勒奖、乔治·波利亚奖，以及美国数学协会颁发的两项莱斯特·佛特奖。关于哈尔莫斯 一些关于数学写作、教学、数学的本质等的“通俗”论文的摘录，见THIS LINK。

J B Conway 在[1]中写到哈尔莫斯 对算子理论的贡献：-

……哈尔莫斯 有许多论文和定理，任何人都会自豪地称其为自己的。但关于他的工作，一直让我印象深刻的是，有数量惊人的主题和问题是当今当前研究中的主导主题，并且都源于他的工作。多年来，哈尔莫斯 展示了一种不可思议的能力，能从给定的数学实体中提取关键性质，并以一种让同事们普遍倾向于进一步观察和探索的方式将其展示在他们面前。

哈尔莫斯 经常访问苏格兰。他定期参加四年一度的圣安德鲁斯学术讨论会。我们第一次见到他是在1972年圣安德鲁斯学术讨论会上，并且完全同意 Alastair Gillespie 在[1]中的评论：-

这些学术讨论会正是哈尔莫斯 所喜欢的那种活动——阐述性数学与娱乐的愉快混合——事实上，就是一个数学假期。

哈尔莫斯在1973年的休假期间有一部分时间是在爱丁堡度过的，并当选为爱丁堡皇家学会会士。他还获得了圣安德鲁斯大学授予的名誉理学博士学位。

2002年，他与妻子哈尔莫斯共同捐赠400万美元，用于重建华盛顿特区的马车房会议中心。他还资助了马车中心的美国数学协会项目。他捐款支持美国数学协会的莱昂哈德·欧拉奖及其面向阐释性写作的哈尔莫斯-莱斯特·佛特奖。哈尔莫斯和Virginia 哈尔莫斯还向美国数学会捐款，设立杜布奖，以奖励杰出的数学阐释性写作。

哈尔莫斯在加利福尼亚州洛斯加托斯因感染肺炎去世，享年九十岁。他的妻子Virginia继续住在洛斯加托斯，并于2015年1月19日在那里去世，享年99岁。Gerald L Alexanderson写道：-

在她生命的最后几周里，她仍在听一本即将出版的关于戈弗雷·哈罗德·哈代的书的草稿，并回忆利特尔伍德（她曾自豪地说，自己曾在三一学院的高桌宴会上坐在他旁边）、法兰克·斯米斯斯以及戈弗雷·哈罗德·哈代时代的其他人。

让我们以哈尔莫斯被问及数学对他意味着什么时的回答来结束这篇传记：-

它是安全。确定性。真理。美。洞见。结构。建筑。我把数学，我称之为数学的那部分人类知识，看作一个东西——一个伟大而辉煌的东西。

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, 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.