I found this book to be a highly interesting biography of a good mathematician and great writer. I say good, because Paul Halmos denies the claim of being a great mathematician in the sense of Paul Cohen or Irving Kaplansky. Rather he says that he is a "professional" mathematician, and this book describes his professional life. But it does it in such a personal way that one cannot help but find it fascinating. I feel that this book not only influenced the way that I think about mathematics, but even the way that I think about life.

A Fantastic Book -- this 400+ page manuscript nicely mixes mathematical science with a historical view of the development (1930's through early 1980's) of mathematical research in the United States. This book is highly readable, extremely enjoyable and quite straightforward with details and opinions. One gets a first hand insight into how the author approached his research, his career, and his life. Halmos has always been a brilliant and skillful writer but his contributions have mostly been in the technical arena; this time he has provided a volume we can all enjoy. I found it difficult to put this book down once I began its reading.

As always, Halmos is an excellent expositor. The brief first chapter has been known to scare away physicists and other intelligent people who don't happen to know ring theory, but fortunately the first chapter can be skipped. The book is at the right level to be used by mathematics graduate students learning Boolean algebras and Stone spaces for the first time.

Some people draw a sharp distinction between the concepts of "Boolean space" (a totally disconnected compact Hausdorff space) and "Stone space", the difference being that a Stone space is the Stone space _of_ a Boolean algebra. A Boolean algebra's Stone space is the space of all of its 2-valued homomorphisms with the topology of pointwise convergence of nets of such homomorphisms. That every Boolean space is the Stone space of some Boolean algebra (namely, the Boolean algebra of all of its clopen subsets) is one of the important facts of "Stone's duality". Halmos never mentions the phrase "Stone space", but he proves the basic facts about "Stone's duality": that the category of Boolean algebras and Boolean homomorphisms is the opposite of the category of Boolean spaces and continuous functions.

The flaw is in Lecture 21. Much of that section is founded upon an error -- that a Boolean algebra may have various non-isomorphic completions, of which one is "minimal". A careful mathematician can reconstruct that lecture and get much that is of value, but some may be unfortunately misled.

One other thing irritates me: Halmos uses the word "non-atomic" rather than the much better term "atomless". The problem with "non-atomic" is that it may be mistaken for "not atomic", and that is a quite different thing.

This book is now out of print. I'd like to see Dover reprint it.

A very good book, going through the stuff in a very nice kind. Giving you intuitive insights, easy proofs, and written with a large sense of humor.

Halmos is one of the great mathematical expositors of the 20th Century, and his book "Finite Dimensional Vector Spaces" stands as the definitive introduction to the subject for budding mathematicians. This book, "Linear Algebra Problem Book", is perhaps best described as an engaging and semi-informal invitation and complement to that original work, which grew out of lectures given by the legendary John von Neumann. In contrast to typical treatments of linear algebra, "Finite Dimensional Vector Spaces" is abstract (introduces determinants through alternating forms), rigorous, concise, and demands a certain level of mathematical maturity. This book, "Linear Algebra" is exactly the opposite. Starting from very little assumed background, it all but gives away the store, written in plain language, anticipating students' questions and misconceptions, and leading them to a deeper understanding of mathematics through the Socratic method. This is not a problem book in the Schaum's outline sense; there is no drill or rote calculations. Every question is carefully chosen to illustrate a point or expose a potential misunderstanding in the student's knowledge or to exercise the student's intuition and ability to make connections. The answers are given as detailed explanations, integral to the exposition, which go far beyond merely answering the questions posed, raising deeper implications and questions. This is an excellent book for beginning students of higher mathematics, and a very user friendly guide to Halmos' classic text.

I'm no mathematician but, I own a number of linear algebra and abstract algebra books and this book makes them all pale in comparison. I really enjoyed working through it. Maybe a bit pedantic for more advanced types but, if you want to get a handle on the concepts this is the book.

This book should have been titled "A Hilbert Space Idea/Problem Book" as it not only challenges the reader to work out interesting problems in operator theory and the geometry of Hilbert space, but also motivates the essential ideas behind these fields. It is definitely a book that, even though out-of-print, will be referred to by many newcomers to operator theory and quantum physics. The insight one gains by the reading of this book is unequaled in any other books in existence on operator theory. It is becoming more rare as mathematics advances, to find books that attempt to explain the intuition behind the abstractions that are manifested in any area of mathematics. The problems in the book deal with both concrete examples and general theorems, and the reader should attempt to try and solve them without looking at the hints. The solutions found by the reader can then be compared with the author's, and some interesting differences will occur.

There are so many interesting discussions in this book that to list them all would probably entail listing everything in the book. The reader will find excellent discussions of the origin of normal operators on infinite dimensional Hilbert spaces as analogs to matrices on finite dimensional spaces; why the weak topology in infinite dimensions is not metrizable; the non-emptiness of the spectrum and why the spectral radius can be computed even though the spectrum cannot; the impossibility of isolated singular operators; the non-continuity of the spectrum: the existence of an operator with a large spectrum and the existence of operators with small spectra in every neighborhood of the large spectrum. The author then goes on to show that the spectrum is an upper semicontinuous function, thus preventing the existence of small spectra arbitrarily close to large spectra. This is an excellent discussion on the meaning and intuition behind semicontinuity; the result that every normal operator is unitarily equivalent to a multiplication and its equivalance to the spectral theorem. The author goes on to explain how one gives up the sigma-finiteness of the measure when doing this, and the origin of functional calculus; the difference between infinite and finite dimensions when attempting a polar decomposition for operators and its connection with partial isometries; the origin of compact operators and their connection with integral equations. The author shows how even the identity operator is not an integral operator on the space of square-integrable functions with Lebesgue measure.

In discussing the spectral theorem in chapter 13 the author states most profoundly: "In some contexts some authors choose to avoid a proof that uses the spectral theorem even if the alternative is longer and more involved. This sort of ritual circumlocution is common to many parts of mathematics; it is the fate of many big theorems to be more honored in evasion than in use. The reason is not just mathematical mischievousness. Often a long but 'elementary' proof gives more insight, and leads to more fruitful generalizations, than a short proof whose brevity is made possible by a powerful but overly specialized tool." In these few sentences the author has characterized the problem with current methods of teaching advanced mathematics. Too often the formalism masks the true meaning and intuitive motivation behind the mathematics. And even though mathematics is being applied to many different areas at an unprecedented rate, pure mathematics seems to be trapped in a local minimum, and I beleive this is due to the reluctance of authors to explain in detail the essentials of their ideas. This book is a perfect example of how mathematics can be taught that requires much thought and creativity on the part of students, without spoon-feeding them and thus encouraging a passive attitude to the learning of mathematics. I salute the author in his achievements in research and in teaching...one can only hope that his approach will be followed in all future works of mathematics.

This book is an overview of measure theory that is somewhat dated in terms of the presentation, but could still be read profitably by someone interested in studying the subject with greater generality than more modern texts. Measure theory has abundant applications, and has even gained importance in recent years in such areas as financial engineering. Those interested in the applications of measure theory to financial engineering should choose another book however, since this one does not even mention the word martingale. After a review of elementary topology and set theory in chapter 1, the author begins to define the elementary notions of measure theory in chapter 2. His approach is more general than other texts, since he works over a ring instead of an algebra. Measures on intervals of real numbers is given as an example. Measures and outer measures are defined, and it is shown how a measure induces an outer measure and how an outer measure induces a measure.

The next chapter explores more carefully the relation between measures and outer measures. It is also shown in this chapter to what extent a measure on a ring can be extended to the generated sigma-ring. The all-important Lebesgue measure is developed here also, and the author exhibits an example of a non-measurable set.

In order to develop an integration theory, one must first characterize the collection of measurable functions, and the author does this in chapter 4. The convergence properties of measurable functions are carefully outlined by the author.

The theory of integration begins in chapter 5, wherein the author follows the standard construction of an integral by first defining integrals over simple functions. Then in chapter 6, signed measures are defined, and the Lebesgue bounded convergence theorem is proven and the Hahn and Jordan decompositions of these measures are discussed. The all-important Radon-Nikodym theorem, which gives an integral representation of an absolutely continuous sigma-finite signed measure, is proven in detail.

One can of course take the Cartesian product of two measurable spaces, and the author shows how to define measures on these products in chapter 7, including infinite products. The physicist reader may want to pay attention to the section on infinite dimensional product measures, as it does have applications to functional integration in quantum field theory (although somewhat weakly).

The author treats measurable transformations in chapter 8, but interestingly, the word "ergodic" is never mentioned. He also introduces briefly the L-p spaces, so very important in many areas of mathematics, and proves the Holder and Minkowski inequalities.

The next chapter is the most important in the book, for it covers the notion of probability on measure spaces. After an brief motivation in the first section of the chapter, probability spaces are defined, and Bayes' theorem is discussed as an exercise. Both the weak and strong law of large numbers is proven in detail.

Things get more abstract in chapter 10, which discusses measure theory on locally compact spaces. Borel and Baire sets on these kinds of spaces are defined, and the author gives detailed arguments on what must be changed when doing measure theory in this more general kind of space.

The book ends with a discussion of measure theory on topological groups via the Haar measure. This chapter also has connections to physics, such as in the Faddeev-Popov volume measure over gauge equivalent classes in quantum field theory. The author does a fine job of characterizing the important properties of the Haar measure.

If you want to stydy measure theory from scratch, I do recommend this book. This book is based on a ring, not an algebra, and is a little old-fashioned. So some people feel uncomfortable. But in particular, product spaces, the Fubini theorem and extension theorems are written very clearly. I'm convinced this book will facilitate your learning in measure theory and probability theory.

This book is an excellent primer on the basics of set theory that all graduate students need, but are not necessarily obtained in the general undergraduate curriculum. Halmos writes in an abbreviated, yet effective style that imparts the necessary details without an excess of words. Theorems and exercises are very few, so it really cannot be used as a textbook. If you need a great deal of explanations, then it is not for you. However, if your need is for a book that distills the essence of set theory down to the shortest possible size, then this book should be yours, either in your college or personal library.

My previous review of Halmos' book stands. Exceptional book, but ... As an example of a question in the book to whet some appetites and in seeking someone's kind mercy in actually answering it for me and putting me out of my misery, consider p.59 on the Axiom of Choice. Quote: if {X (sub)i} is a finite sequence of sets, for i in n say, then a necessary and sufficient condition that their Cartesian product be empty is that at least one of them be empty ... (The case n=0 leads to a slippery argument about the empty function; the uninterested reader may start his induction at 1 instead of 0). Unquote. Induction from 1 is no problem. The slippery argument stuff (and other similar pearls thoughout the book) sends me wild. What is the slippery argument. Please. Anyone. With thanks to Paul Halmos for making my life 'miserably interesting' (sic)!!

An exceptional book. The book, however, has little pedagogical value. I would not recommend those starting out in mathematics to purchase it. It is definitely for the mathematically mature. Indeed, it is the type of book that may force some to consider abandoning mathematics if it is read without assistance too early in their development. The lack of answers to exercises amplifies these considerations when the book is used for self study as there are few means to understand whether one is on the right track, especially when the less natural approach of recursion is required to answer some questions. If you want to maximise your understanding of set theory, however, this is an essential book to get. In a sense, when read in conjunction with Paul Halmos' background and some quotes attributed to him found elsewhere on the Internet, the book is almost autobiographical.

This book reviews some ideas Halmos worked on in the 1950s: the algebraization of predicate logic. The result was polyadic algebra, which has been unfairly neglected since. Tarski, Henkin, and their Berkeley students worked on a rival research program that culminated in the better known cylindric algebras. The treatment remains at the undergrad level, because Halmos stops short short of polyadic predicates. Halmos's "Algebraic Logic," which AMS keeps in print and is a fine read, contains all of Halmos's professional writings on polyadic algebra.

While Halmos does not cover all of first order logic, he does an excellent job of introducing the reader to the great power and depth of Boolean algebra, revealed by Marshall Stone and Tarski in the 1930s, and other Poles in the 1950s. By this I mean Boolean algebra coupled with the notions of filters, ideals, generators, and quotient algebras. The metatheory of the propositional calculus has a very elegant Boolean representation.
For that matter, the completeness of first order logic has a nice polyadic algebra translation.

Lattice theory is an extremely powerful generalization of Boolean algebra that has not attracted the attention it deserves. If Halmos had written a text on lattice theory, that situation would in all likelihood have ended. Halmos and Givant include an all-too-brief tantalizing chapter on lattices.

If this book has a drawback, it is the relative unsophistication of its first 40 odd pages, an introduction to logic. This is especially disappointing given that Givant is a logician, and an excellent one at that, being a student of Tarski's.

The books main asset is Halmos's lively prose style, unparalleled in modern mathematics. Math PhD students should study this book closely as a superb example of how to exposit mathematics.

It can be strongly argued that logic is the most ancient of all the mathematical sub-disciplines. When mathematics as we know it was being created so many years ago, it was necessary for the concepts of rigid analytical reasoning to be developed. Of the three earliest areas, geometry was born out of the necessity of accurately measuring land plots and large buildings and number theory was required for sophisticated counting techniques. Logic, the third area, had no "practical" godfather, other than being the foundation for rigorous reasoning in the other two. In the intervening years, so many additional areas of mathematics have been developed, with logic and logical reasoning continuing to be the fundamental building block of them all. Therefore, every mathematician should have some exposure to logic, with the simple history lesson automatically being included. This short, but excellent book fills that niche.
The title accurately sets the theme for the entire book. Algebra is nothing more than a precise notation in combination with a rigorous set of rules of behavior. When logic is approached in that way, it becomes much easier to understand and apply. This is especially necessary in the modern world where computing is so ubiquitous. Many areas of mathematics are incorporated into the computer science major, but none is more widely used than logic. Written at a level that can be comprehended by anyone in either a computer science or mathematics major, it can be used as a textbook in any course targeted at these audiences.
The topics covered are standard although the algebraic approach makes it unique. One simple chapter subheading, 'Language As An Algebra', succinctly describes the theme. Propositional calculus, Boolean algebra, lattices and predicate calculus are the main areas examined. While the treatment is short, it is thorough, providing all necessary details for a sound foundation in the subject. While the word "readable" is sometimes overused in describing books, it can be used here without hesitation.
Sometimes neglected as an area of study in their curricula, logic is an essential part of all mathematics and computer training, whether formal or informal. The authors use a relatively small number of pages to present an extensive amount of knowledge in an easily understandable way. I strongly recommend this book.

Published in Smarandache Notions Journal reprinted with permission.

In his "automathography" Halmos described his views on logic which he had in the 1960's. He felt that logic, as usually stated, was very un-profound, unrigorous, combinatorial amusement. He felt additionally that logic could be put on a firm algebraic footing through the theory of Boolean rings. At that time he interpreted many things in logic in terms of Boolean rings. This book is, in some sense, the child of these labors. Halmos created this book in his usual easy to read style, and when he said that few prerequisites were assumed, he meant it. I found the (short) book very interesting, but I also found the introductory pages seemed to drag. Perhaps this is because I already know something about logic, but the rest of the book was interesting and self contained. This book was lighter than most logic books I've seen. These books were mainly written by philosophers in some capacity or other, and they never stopped their thick prose.

Have you ever wanted to be a mathematician? Read this book to see what it's like.

Borders on unreadable. It's a joke that respected math professors and textbook reviewers all think of this book as a classic. Although widely praised when it first came out back in the 1940's, you have to remember that it was also the first book on the subject published in English. Not at all recommended for students approaching this material for the first time.

This book is the best if you are looking for an abstract approach to linear algebra. It provides elegant proofs to theorems that usually seem long-winded and awkward (like the cauchy-schwarz inequality). Sometimes in your lectures you may get to the point thinking "can't this be proven more elegant?" and you simply open halmos and it is there.

Note that this book does not deal alot with matrices, everything of the theory is there, but you might miss illustrations and applications. In this case I recommend to back it up with Gilbert Strangs Linear Algebra and its Applications, which has an intuitive, matrice-oriented approach.

Considering the price and the wide range of topics often left out in other books (like Nilpotence, Jordanform, Spectral Theorem,...) this simply is the one book you should buy and keep for reference.

Halmos always exemplifies clarity in writing, but sometimes only for those who either think like mathematicians or are working on learning how to do so. Others should stay away, and stop blaming Halmos if their instructors inappropriately prescribe this book for students for whom it is not suitable.