For extracts from reviews of Naive set theory see THIS LINK.
Since Halmos wrote a large number of exceptional texts, we have split this collection into two pieces. Below is the second half of our collection. For the first half of the collection of Halmos's books, see THIS LINK.
9. A Hilbert space problem book (1967), by Paul R Halmos.
Mathematical Reviews MR0208368 (34 #8178).
The book is divided into three parts: problems, hints and solutions. According to the instructions, if one is unable to solve a problem, even with the hint, he should, at least temporarily, grant it (if it is a statement) and proceed to another. He should be prepared, however, to use the as yet unproved statement in the hope of thereby getting some clues as to its solution. If he does solve a problem he should look at the hint and the solution anyway for perhaps some other variations on the theme. The rules are simple and the advantages of following them to the conscientious and diligent reader are surely obvious and incalculable. Nevertheless, there is the ever lurking temptation to combine each problem-hint-solution into a unified whole, to be read together and at one time. Although this procedure admittedly thwarts, at least partially, the aim of the book, it is certainly not without its own rewards. Thus, a researcher thereby may quickly find a compact presentation of a particular issue, often including some history, references, and enlightening side remarks.
Amer. Math. Monthly 91 (9) (1984), 592-594.
A recurring theme is the perturbation of an operator by a compact operator, beginning with the Fredholm alternative and a 1909 result of Weyl on the invariance of the limit points of the spectrum under such perturbations. ... Because of the diversity of topics that it includes, single operator theory is ideally suited to a book of the nature of Halmos's (we can imagine a Homotopy Theory Problem Book being less successful). The second edition contains 250 problems as against 199 in the first. It remains an excellent, compact survey of single operator theory, as well as a valuable source for Hilbert space techniques. The form, content, and style of the second edition deserve the same high praise as the first.
10.2. Review by: Philip Maher.
The Mathematical Gazette 73 (465) (1989), 259-260.
It might seem somewhat bizarre to be reviewing here a book that was first published in 1967 and one, moreover, whose subject matter is at the upper end of accessibility for the Gazette readership. Yet Halmos's 'A Hilbert space problem book' has never been reviewed in these pages before; the second edition (under review) differs quite a bit from the first; and, of course, anything by Halmos, however technical, promises to be lucid and entertaining. In fact, this book - with its well-nigh unparalleled form - is an object lesson in the communication of mathematics. For it does not follow that boring and so-often unilluminating format of definition, lemma, theorem, proof (a format which seems virtually de-rigeur for the exposition of advanced mathematics). Rather, this book consists of problems; some 250 of them (and their solutions) which cover much of the theory of (single) linear operators on Hilbert space. Of course, I am aware that other books have appeared that approach parts of mathematics in a problem-orientated (as distinct from, so to speak, a subject-orientated) way: one thinks of Burns's 'A pathway to number theory' for example. But Halmos's books is the first to do this at an advanced (rather than at an elementary) level.
Mathematical Reviews MR0517709 (80g:47036).
The book contains a very valuable amount of information on integral operators. The theory is richly illustrated by examples and counterexamples. Many open problems are discussed. A large number of them have been solved recently by Schep and W Schachermayer, and the solutions will appear soon in print. Also, the history of the subject is well presented. The style of the book is lucid, lively and entertaining, as we have come to expect from the senior author.
11.2. Review by: Adriaan Zaanen.
Bull. Amer. Math. Soc. (N.S.) 1 (6) (1979), 953-960.
The Halmos-Sunder book is written in a lively style, as was to be expected. It is a mine of information for any analyst interested in operators, in particular operators on L2spaces. The theory is illustrated by examples as well as by counterexamples; open problems are mentioned and sometimes analyzed. A list of bibliographical notes gives information about the history of the subject. The preface ends with the remark that the book contains only a part of a large subject, with only one of several approaches, and with explicit mention of only a few of the many challenging problems that are still open. I agree, but I hope that nevertheless it will be clear from my comments in this review that I believe the present contribution to operator theory by Halmos and Sunder is a valuable one.
The Mathematical Gazette 70 (453) (1986), 253-255.
P R Halmos has a well-deserved reputation as an expositor of mathematics. One of his favourite techniques is to excavate for the simplicity underlying layers of complexity, to extract it and to display it in a strikingly illuminating way. He also of course has made important contributions to mathematics per se, but on his own self-assessment his research achievements rank only fourth, behind writing, editing and teaching. These abilities as teacher of and doer of mathematics, combined with mild but significant eccentricity has made him a 'name' in mathematics, a name about whom the mathematical public will be sufficiently curious to guarantee interest in his autobiography. ... this is a fascinating addition to recent mathematical culture by one of its makers. The main message I absorbed from it was a set of conditions required for success in mathematics: talent, yes; single-mindedness, almost as obvious; sense of humour, essential when the going gets tough; and love, yes that is the right word - you must love mathematics, and that means all the ingredients, passion, pain and loyalty.
12.2. Review by: Henry Helson.
Mathematical Reviews MR0789980 (86m:01059).
This autobiography is a frank, personal, witty commentary on mathematicians and mathematics by one of the most influential, and observant, mathematicians of our time. It is much more about the profession of mathematics than about the personal life of its author. ... He makes two main points. The first is the importance of being literate. The ability to speak and write effectively, preferably in more than one language, is essential to effectiveness in all professional activity. And second, a real professional must work in all aspects of the job: research of course, but also teaching in several formats, exposition at several levels, refereeing and editing, departmental chores, and participation in meetings and conferences. His standard is high.
12.3. Review by: John A Dossey.
The Mathematics Teacher 79 (6) (1986), 481-482.
This autobiographical sketch details the professional aspects of the career of the distinguished American mathematician Paul Halmos. It gives the reader a delightfully candid view of the evolution of his career of research, teaching, travel, editing, and service from his secondary school days to the present. ... The book is exciting, witty, and well worth the time invested in its study. It communicates what it means to be a mathematician.
12.4. Review by: Gian-Carlo Rota.
Amer. Math. Monthly 94 (7) (1987), 700-702.
Every mathematician will rank other mathematicians in linear order according to their past accomplishments, while he rates himself on the promise of his future publications. Unlike most mathematicians, Halmos has taken the unusual step of printing the results of his lifelong ratings. By and large, he is fair to everyone he includes in his lists (from first rate (Hilbert) to fifth rate (almost everybody else)), except towards himself, to whom he is merciless (even in the choice of a title to the book: "I want to be a mathematician," as if there were any question in anybody's mind as to his professional qualifications). ... The leading thread of his exposition, what makes his narration entertaining (rather than just interesting), is mathematical gossip, which is freely allowed to unfold in accordance to its mysterious logic. The reader will be thankful for being spared the nauseating personal details that make most biographies into painful reading experiences ... Whatever does not relate to the world of mathematics is ruthlessly and justly left out (we hardly even learn whether he has a wife and kids).
Instead, the book is about his life as a mathematician among mathematicians. It shows us a little about how Halmos thinks about mathematics, about what interested and motivated him, and about how he interacted with others. It includes a lot of what might, somewhat uncharitably, be described as "gossip": stories and anecdotes about mathematics, mathematics departments, and mathematicians. In my experience, mathematicians love this sort of thing. Those of my colleagues who have read this book have enjoyed it. My students have liked it much less, partly because they aren't that interested in the world of mathematics, partly because they feel "turned off" by what they describe as Halmos's "arrogance." I think what bothers them is Halmos's bluntness about what counts and what does not count as significant in mathematics. That Halmos's harshness is mostly directed at his own work doesn't change my students' assessment. If all this famous guy can say, after trying for fifty years, is "I want to be a mathematician," they argue, then we students have no chance at all.
Would you look at some of the snapshots I have taken in the last 40-odd years? If you put a penny into a piggy bank every day for 45 years, you'll end up with something over 16,000 pennies. I have been a snapshot addict for more than 45 years, and I have averaged one snapshot a day. Over a third of the pictures so accumulated have to do with the mathematical world: they are pictures of mathematicians, their spouses, their brothers and sisters and other relatives, their offices, their dogs, and their carillon towers. The pictures were taken at the universities where I worked, and the places where I was a visitor (for a day or for a year), and, as you will see, many of them were taken over food and drink. That's rather natural, if you think about it. It is not easy, and often just not possible, to snap mathematicians at work (in a professional conversation, thinking, lecturing, reading) - it is much easier to catch them at tea or at dinner or in a bar. In any event, the result of my hobby was a collection of approximately 6000 "mathematical" pictures, and when it occurred to me to share them with the world I faced an extremely difficult problem of choice. ... The people included are not necessarily the greatest mathematicians or the best known. If I think a picture is striking, or interesting, or informative, or nostalgic, then it is here, even if the theorems its subject has proved are of less mathematical depth than those of a colleague whose office is two doors down the hall.
14.2. Review by: Arthur M Hobbs.
Mathematical Reviews MR0934204 (89f:01067).
This is a book of considerable merit in several different ways. Beginning with the most frivolous use, it is very nearly the perfect book for a mathematician's coffee table (it only lacks colour). Less trivially, with 604 photographs of (mostly) mathematicians and two photographs of the old and new buildings at Oberwolfach, each picture accompanied by an informative caption, the book is pleasant reading for any mathematician, for many mathematician's spouses, and for any other person interested in mathematics as it is lived. ... as a historical document, this book will be valuable in at least two ways. It gives a fascinating cross-section of mathematical life in the mid-twentieth century, and it provides considerable insight into the personality and interests of Professor Halmos himself.
Amer. Math. Monthly 99 (9) (1992), 888-890.
What makes a problem interesting? Its statement should be simple, not requiring excessive explanations, and the solution should be readily understandable by the intended audience. [In] Halmos's book ... his problems ... meet these criteria admirably. ... Halmos has won several writing awards, and the reader won't be disappointed in the prose with which he wraps the problems ...The book lives up to its title, which promises problems for both young and old. But is it the young or the old who are more likely to be impressed by the pretty and elegant elementary problems? I'm not sure. In any event, for the experienced mathematician or beginning graduate student seeking meatier fare the book contains several chapters with problems for the more mature reader.
15.2. Review by: Lionel Garrison.
The Mathematics Teacher 85 (7) (1992), 592.
Buy this book. In fact, buy several. Give them to your students and colleagues, and save one for when your first copy wears out. ... [this book], impels the reader to get out a pencil and start doing mathematics. Paul Halmos, one of the pre eminent mathematicians and teachers of our time, here shares his personal treasury of favourite problems. Perhaps a third of these 165 problems are at least comprehensible to able high school students who have studied calculus and probability and who enjoy being challenged. The remainder generally assume the standard undergraduate pro gram in analysis, topology, and abstract algebra.
15.3. Review by: G A Heuer.
Mathematical Reviews MR1143283 (92j:00009).
The 165 problems in this charming book are classified roughly by the mathematical subdiscipline into which they most naturally fit. ... While a good many of the problems are catchy enough to prick the interest of most readers without further help, most are preceded by the kind of skilful discussion for which the author is so well known, to make the problem more appealing and to help make clear why this is exactly the natural question to pose in this context. The title is apt: there is much here for the veteran professional mathematician; there is also a good bit for the budding mathematician who is still a high school pupil.
The Mathematical Gazette 81 (490) (1997), 168-170.
... it is the quality of Halmos' commentary that made [the book] such a treat to review and which means that I can recommend it without hesitation to all lecturers (who think they can teach linear algebra) and all sufficiently mature students (who think they have learnt it). His quiet emphasis on details, his penetrating insights into what makes an approach to a proof plausible, or the mode of construction of a counter-example clear, and his 'heart-on-sleeve' approach to problem-posing, in explaining why he cast the problems in the form he did, are matchless: you feel you are in the company of a master expositor, striding together through the world of linear algebra while he points out the flora, fauna, topography and the way ahead. [It is] a book bristling with lovely touches ...
16.2. Review by: Robert Messer.
Amer. Math. Monthly 105 (6) (1998), 577-579.
For students going on in mathematics, linear algebra serves as a transition to upper-level mathematics courses. In addition to learning the subject matter of linear algebra itself, these students must be fortified with a degree of mathematical maturity in working with axioms and definitions, basic proof techniques, and mathematical terminology and notation. These issues cannot be left to chance; they must be addressed explicitly to prepare students for courses such as abstract algebra and real analysis. As a textbook for a linear algebra course, Paul Halmos's 'Linear Algebra Problem Book' satisfies these criteria. ... The conversational style of writing in this book occasionally lapses into annoying chattiness. A definition can be guessable and an answer conjecturable. A corollary can be unsurprising or minute but enchanting. Within three sentences the zero linear functional has two symbols and goes from most trivial to most important and ends up uninteresting.
16.3. Review by: Jaroslav Zemánek.
Mathematical Reviews MR1310775 (96e:15001).
Understanding simple things such as basic linear algebra does not seem to be an easy task. Indeed, the author offers original insights illuminating the essence of the associative and distributive laws, and the underlying algebraic structures (groups, fields, vector spaces). The core of the book is, of course, the study of linear transformations on finite-dimensional spaces. The problems are intended for the beginner, but some of them may challenge even an expert. ... Needless to say, more emphasis on the history of the subject would be attractive in a book of this type. In brief, the reviewer regrets that the author chose not to go deeper into the subject; he shows a few trees, but makes little attempt to see the forest.
The Journal of Symbolic Logic 63 (4) (1998), 1604.
This slim book provides an introduction to logic with the goal, as suggested by the title, of demonstrating that logic can be profitably understood from an algebraic viewpoint. Instead of making the point by providing the reader an extensive treatment of algebraic logic, the book takes a modest concrete approach by limiting its consideration to that part of logic most familiar to a general audience, namely the propositional calculus and the monadic predicate calculus. The presentation reflects Halmos's pedagogical style. Topics are introduced first by logical considerations, then the ideas are abstracted, and finally they are placed in an appropriate general algebraic context. The book reads like an essay. This is not to say the presentation is not rigorous. It is. Details are presented when necessary to convey the ideas, but without overwhelming readers. The presentation is crisp and lucid yet informal. It is as if the principles of logic are being explained over a cup of coffee. The book is directed towards a general (mathematically literate) audience with an interest in modern logic. Nevertheless, the prerequisite of "a working knowledge of the basic mathematical notions that are studied in a first course in abstract algebra" should be taken seriously. It is not a textbook (in the usual sense) even though it is based on notes from a course in logic by Paul Halmos.
17.3. Review by: Graham Hoare.
The Mathematical Gazette 84 (499) (2000), 172-173.
[The book] is intended 'to show that logic can (and perhaps should) be viewed from an algebraic perspective .... Moreover, the connection between the principal theorems of the subject and well-known theorems in algebra become clearer.' Readers anticipating arguments based on truth tables or diagrams of switching circuits will be disappointed. In compensation they will be entertained by a rich array of algebraic concepts such as prime and maximal ideals, filters, homomorphisms, equivalence classes, kernels, quotient algebras and duality, all in the service of logic. As the authors state, 'propositional logic and monadic predicate calculus - predicate logic with a single quantifier, are the principal topics treated'. ... the whole will serve as a neat, succinct, introduction to logic particularly for readers very much at home with algebraic concepts.
17.4. Review by: Myra R Lipman.
The Mathematics Teacher 92 (4) (1999), 371.
The authors of this book have targeted a wide-ranging audience; however, the book cannot be all things to all people. ... Because of the scope and depth of material, this book would be most useful as a classroom reference or supplement. If the book included many more examples and some exercises, it would be outstanding and certainly more helpful to students. Viewing logic from an algebraic perspective is an intriguing concept, and the authors succeed in giving a concise overview of relatively complex material in an intelligible manner.
17.5. Review by: Natasha Dobrinen.
The Bulletin of Symbolic Logic 16 (2) (2010), 281-282.
This is an excellent and much-needed comprehensive undergraduate textbook on Boolean algebras. It contains a complete and thorough introduction to the fundamental theory of Boolean algebras. Aimed at undergraduate mathematics students, the book is, in the first authors words, "a substantially revised version of Paul Halmos' Lectures on Boolean algebras." It certainly achieves its stated goal of "steering a middle course between the elementary arithmetic aspects of the subject" and "the deeper mathematical aspects of the theory" of Boolean algebras.
17.6. Review by: Marcel Guillaume.
Mathematical Reviews MR1612588 (99m:03001).
... this booklet is a gem, whose reading the reviewer ... highly recommends, fundamentally because it gives a better designed, direct and concise overview of logic, going beyond the topics explicitly treated, which are in fact reduced just to the minimum needed in order to explain and introduce the key notions, and to explain and prove fundamental theorems in the special cases considered. As to the form, the style is vivid and clear, using simple words, and free of long and complex technicalities. The text is rich in brief comments explaining the ideas behind the reasoning and calculations, and frequently refers to simple examples and to the basic notions of universal algebra.