## Reviews of Paul Halmos' books Part 1

For extracts from reviews of Since Halmos wrote a large number of exceptional texts, we have split this collection into two pieces. For the second half of the collection of Halmos's books, see THIS LINK. |

**1. Finite dimensional vector spaces (1942), by Paul R Halmos.**

**1.1. Review by: E R Lorch.**

*Mathematical Reviews*MR0033869

**(11,504d).**

A purpose of this study is to present n-dimensional transformation theory from the abstract point of view. Normally, due to the existence of a finite basis, elementary matrix theory has an aspect not at all suggesting that the infinite dimensional theory of operators is a natural extension of it. The author exploits as completely as possible the methods and notions of the infinite in his presentation of the finite case; such a program has long been needed.

**1.2. Review by: Mark Kac.**

*Bull. Amer. Math. Soc.* **49** (5) (1943), 349-350.

In this book the author presents the topics covered usually in an introductory course in algebra (matrices, linear equations, linear transformations, and so on) from the point of view of a modern analyst interested in general vector spaces. The ever-growing interest in Hilbert and more general linear spaces makes the appearance of the book very timely, especially since it furnishes an excellent introduction to the subject certainly within the grasp of a first-year graduate student or even a good senior or junior. The topics are treated in such a manner as to make future generalizations look both natural and suggestive. This sometimes is done at the expense of the shortness of exposition. Some theorems, as the author himself confesses, could be proved in fewer lines. He prefers, however, longer proofs that admit a generalization to shorter ones that do not.

**2. Finite dimensional vector spaces, 2nd Edition (1958), by Paul R Halmos.**

**2.1. From the Preface.**

My purpose in this book is to treat linear transformations on finite- dimensional vector spaces by the methods of more general theories. The idea is to emphasize the simple geometric notions common to many parts of mathematics and its applications, and to do so in a language that gives away the trade secrets and tells the student what is in the back of the minds of people proving theorems about integral equations and Hilbert spaces. The reader does not, however, have to share my prejudiced motivation. Except for an occasional reference to undergraduate mathematics the book is self-contained and may be read by anyone who is trying to get a feeling for the linear problems usually discussed in courses on matrix theory or "higher" algebra. The algebraic, coordinate-free methods do not lose power and elegance by specialization to a finite number of dimensions, and they are, in my belief, as elementary as the classical coordinatized treatment. I originally intended this book to contain a theorem if and only if an infinite-dimensional generalization of it already exists. The tempting easiness of some essentially finite-dimensional notions and results was, however, irresistible, and in the final result my initial intentions are just barely visible. They are most clearly seen in the emphasis, throughout, on generalizable methods instead of sharpest possible results. The reader may sometimes see some obvious way of shortening the proofs I give. In such cases the chances are that the infinite-dimensional analogue of the shorter proof is either much longer or else non-existent.

**2.2. Review by: J L B Cooper.**

*The Mathematical Gazette* **44** (348) (1960), 142-143.

The purpose of this book is to give an account of the theory of linear operators and manifolds in finite-dimensional spaces suitable as an introduction to the theory of operators in Hilbert Space. It adopts an approach based on operator theory, makes little use of matrices and determinants, though these are mentioned, and gives some indication of the use of transcendental methods, such as Zorn's lemma. After an account of operators in general finite spaces, including a rather sketchy account of the Jordan canonical form, the scalar product spaces and the decomposition of hermitian and normal operators in these are treated with great thoroughness. Finally convergence of operators, with an elementary ergodic theorem, are studied, and a brief account of Hilbert Space follows. The book is written clearly and carefully, and has numerous examples well chosen to illustrate its point of view. It can be recommended strongly for the student of its subject.

**2.3. Review by: Albert Wilansky.**

*Amer. Math. Monthly* **66** (6) (1959), 528-529.

Foreign mathematicians are warned not to search in dictionaries for zeroish, zeroness, askable, .... In style, Professor Halmos follows G H Hardy in the role of a "missionary preaching to the cannibals."

**3. Measure theory (1950), by Paul R Halmos.**

**3.1. From the Preface.**

My main purpose in this book is to present a unified treatment of that part of measure theory which in recent years has shown itself to be most useful for its applications in modern analysis. If I have accomplished my purpose, then the book should be found usable both as a text for students and as a source of reference for the more advanced mathematician. I have tried to keep to a minimum the amount of new and unusual terminology and notation. In the few places where my nomenclature differs from that in the existing literature of measure theory, I was motivated by an attempt to harmonize with the usage of other parts of mathematics. There are, for instance, sound algebraic reasons (or using the terms "lattice" and "ring" for certain classes of sets - reasons which are more cogent than the similarities that caused Hausdorff to use "ring" and "field." It might appear inconsistent that, in the text, many elementary notions are treated in great detail, while, in the exercises, some quite refined and profound matters (topological spaces, transfinite numbers, Banach spaces, etc.) are assumed to be known. The material is arranged, however, so that when a beginning student comes to an exercise which uses terms not defined in this book he may simply omit it without loss of continuity. The more advanced reader, on the other hand, might be pleased at the interplay between measure theory and other parts of mathematics which it is the purpose of such exercises to exhibit.

**3.2. Review by: J L B Cooper.**

*The Mathematical Gazette* **35** (312) (1951), 142.

[The book] gives a comprehensive account of those aspects of the theory of measure and integration which are important in general measure spaces and in topological spaces and groups. ... The exposition throughout is masterly. Proofs are clear, precise and elegant. Care has been taken to avoid the indigestion which accompanies the reading of abstract theories, by discussions in each section aimed at giving the reader an intuitive grasp of the subject and an idea where he is going, and by well-chosen sets of examples. These last should not be overlooked by the reader: they contain not only helpful illustrative matter on the main text, but also accounts of further developments of the subject.

**3.3. Review by: Harry M Gehman.**

*Mathematics Magazine* **26** (3) (1953), 173-174.

[The book] is intended for use as a textbook in graduate courses in Measure Theory. It should also prove useful for reference purposes with its collection of theorems, its well chosen problems, and its penetrating comments on the subjects which it touches. ... Unlike some textbooks, the exercises in Halmos' book are not trivial. They contain well chosen special cases to illustrate the theory, alternate methods of proofs of theorems, and additional definitions and theorems to extend the material of the text.

**3.4. Review by: S Kakutani.**

*Mathematical Reviews* MR0033869 **(11,504d).**

This book presents a unified theory of the general theory of measure and is intended to serve both as a text book for students and as a reference book for advanced mathematicians. This book is written in a very clear style and will make an excellent text book for those graduate students who are already familiar with the theory of Lebesgue integration in a Euclidean space. But the treatment of the subject is rather abstract so that the book is perhaps not to be recommended for beginners. On the other hand, this book contains many new results obtained in the last ten years, and will be a very useful source of reference for research mathematicians.

**3.5. Review by: J C Oxtoby.**

*Bull. Amer. Math. Soc.* **59** (1) (1953), 89-91.

In this book Professor Halmos presents an account of the modern theory of measure and integration in the generality required for the study of measure in groups. Thus finiteness conditions are imposed only where necessary, and algebraic and topological aspects are appropriately stressed. Although written primarily for the student, the many novel ideas in the book and its store of interesting examples and counter examples have already made it an indispensable reference for the specialist. The clarity of expression and the sprightly style which are characteristic of the author make the exposition a pleasure to follow. ... It seems likely that this book will come to be recognized as one of the few really good text books at its level. It can hardly fail to exert a stimulating influence on the development of measure theory.

**4. Introduction to Hilbert space and theory of spectral multiplicity (1951), by Paul R Halmos.**

**4.1. Review by: J L B Cooper.**

*The Mathematical Gazette*

**36**(317) (1952), 218-219.

The main purpose of this book is to make available in English the theory of the unitary invariants of normal operators in Hilbert Space, and the last of its three chapters is devoted to this subject. The first two contain an excellently set out account of Hilbert Space and of the bounded operators, projectors and bounded hermitian and normal operators on it. The development of the theory of these last operators is made to rest on measure theory: as a result, and by making full use of the latest work on the subject, the author gives a treatment in which the geometry of the space plays a large part. ... The book can be recommended as an introduction to Hilbert Space theory, for a certain class of readers. Its merits are its accuracy, its concise and lively style and its concern to give the reader an idea of the general direction of the arguments. However, it is clearly written for readers with a training in the abstract modes of mathematical reasoning and, particularly, in measure theory. It would not be very suitable for readers interested in physical applications of the theory, partly because these applications involve only separable spaces, but mainly because no mention is made of unbounded operators and consequently of differential operators ... These minor points should not obscure the fact that Professor Halmos has written a stimulating and useful book

**4.2. Review by: B Sz-Nagy.**

*Mathematical Reviews* MR0045309** (13,563a)**.

The main purpose of this book is to present the so-called multiplicity theory and the theory of unitary equivalence, for arbitrary spectral measures, in separable or not separable Hilbert space. This is developed in chapter III; the preceding parts serve as an introduction.

**4.3. Review by: E R Lorch.**

*Bull. Amer. Math. Soc.* **58** (3) (1952), 412-415.

There is little doubt that the author of this book enjoyed himself thoroughly during its preparation. Reading the result afforded this reviewer considerable pleasure. In one hundred and nine well-packed pages one finds an exposition which is always fresh, proofs which are sophisticated, and a choice of subject matter which is certainly timely. Some of the vineyard workers will say that P R Halmos has become addicted to the delights of writing expository tracts. Judging from recent results one can only wish him continued indulgence in this attractive vice. The present work may confidently be recommended. However, beginners in the field should be cautioned before they rush off to secure a copy. Unless one is equipped and in training, one should not attempt the expedition. One must not be misled by the title. For this introduction to Hilbert space, one has to be an expert in measure theory. As a matter of fact it is best to have read the author's book on measure theory or its equivalent. ... Most pages exhibit a zest for play as well as work which is refreshing. Indeed, at times one may have a vague apprehension that the author is preparing a prank or baiting a trap; however it seldom turns out to be more than a friendly tweak given with a wink. Such an intimate style, in the present desert of works written with an unexceptionable scientific detachment, is warmly welcome. It is certainly a facet to the general success enjoyed by Halmos, previous books.

**5. Lectures on ergodic theory (1956), by Paul R Halmos.**

**5.1. Review by: Yael N Dowker.**

*Mathematical Reviews*MR0097489

**(20 #3958)**.

This book is the first work on ergodic theory in book form since E Hopf's 'Ergodentheorie' appeared on 1937. Its contents are based on a course of lectures given by the author at the University of Chicago in 1956. The first of these facts makes the book very welcome; more so since the book is written in the pleasant, relaxed and clear style usually associated with the author. The material is organised very well and painlessly presented. A number of remarks, ranging from the serious to the whimsical, add insight and enjoyment to the reading of the book.

**5.2. Review by: Yael N Dowker.**

*Bull. Amer. Math. Soc. ***65** (4) (1959), 253-254.

The author, in the apology (preface) to the book, asks the reader to regard these notes as "designed to rekindle" interest in the subject. From this point of view and considering the excellent and effortless style of the book it is doubly regretful that the material discussed is so restricted in time and person. There is almost no indication of work done during the last decade, and the reviewer cannot but be disappointed that the reader is left unaware of the recent sparks of interest found by workers in such branches of ergodic theory as, for instance, those related to probability theory, number theory, abstract ergodic theory, dynamical systems in general and geodesic flows in particular.

**6. Entropy in ergodic theory (1959), by Paul R Halmos.**

**6.1. From the Preface.**

Shannon's theory of information appeared on the mathematical scene in 1948; in 1958 Kolmogorov applied the new subject to solve some relatively old problems of ergodic theory. Neither the general theory nor its special application is as well known among mathematicians as they both deserve to be; the reason, probably, is faulty communication. Most extant expositions of information theory are designed to make the subject palatable to non-mathematicians, with the result that they are full of words like "source" and "alphabet". Such words are presumed to be an aid to intuition; for the serious student, however, who is anxious to get at the root of the matter, they are more likely to be confusing than helpful. As for the recent ergodic application of the theory, the communication trouble there is that so far the work of Kolmogorov and his school exists in Doklady abstracts only, in Russian only. The purpose of these notes is to present a stop-gap exposition of some of the general theory and some of its applications. While a few of the proofs may appear slightly different from the corresponding ones in the literature, no claim is made for the novelty of the results. As a prerequisite, some familiarity with the ideas of the general, theory of measure is assumed; Halmos's *Measure theory* (1950) is an adequate reference.

**7. Algebraic logic (1962), by Paul R Halmos.**

**7.1. From the Introduction.**

It has often happened that a theory designed originally as a tool for the study of a physical problem came subsequently to have purely mathematical interest. When that happens, the theory is usually generalized way beyond the point needed for applications, the generalizations make contact with other theories (frequently in completely unexpected directions) and the subject becomes established as a new part of pure mathematics. The part of pure mathematics so created does not (and need not) pretend to solve the physical problem from which it arises; it must stand or fall on its own merits. Physics is not the only external source of mathematical theories; other disciplines (such as economics and biology) can play a similar role. A recent (and possibly somewhat surprising) addition to the collection of mathematical catalysts is formal logic; the branch of pure mathematics that it has precipitated will here be called *algebraic logic. *Algebraic logic starts from certain special logical considerations, abstracts from them, places them into a general algebraic context, and, *via *the generalization, makes contact with other branches of mathematics (such as topology and functional analysis). It cannot be overemphasized that algebraic logic is more algebra than logic. Algebraic logic does not claim to solve any of the vexing foundation problems that sometimes occupy logicians. All that is claimed for it is that it is a part of pure mathematics in which the concepts that constitute the skeleton of modern symbolic logic can be discussed in algebraic language. The discussion serves to illuminate and clarify those concepts and to indicate their connection with ordinary mathematics. Whether the subject as a whole will come to be considered sufficiently interesting and sufficiently deep to occupy a place among pure mathematical theories remains to be seen.

**7.2. Review by: Donald Monk.**

*Amer. Math. Monthly* **71** (6) (1964), 708-709.

By algebraic logic the author means that branch of general algebra which deals with algebraic structures mirroring in some sense certain formal logics. ... As Halmos indicates, one of the main problems in algebraic logic is to state and prove algebraically various important theorems of logic. ... Halmos' book is highly recommended as an introduction for those who wish to study logic from a purely algebraic point of view.

**7.3. Review by: Leon Henkin.**

*Science, New Series* **138** (3543) (1962), 886-887.

Hopefully, the fragmentation of science caused by increasing specialization will be counterbalanced by the development of new modes of unification. In the contemporary development of mathematics one such unifying influence is the massive intrusion of algebraic concepts and methods into all mathematical fields. Mathematical logic has been intertwined with algebra from its beginnings, through Boole's discovery that simple laws of logic can be expressed symbolically as algebraic equations. But only in very recent years has the algebraic viewpoint in logic been systematized to the point where an almost complete account of logic can be given in algebraic terms. Alfred Tarski pioneered this enterprise through successive exploration of Boolean, relation, and cylindric algebras. The most detailed contributions are found in the papers of Paul Halmos on polyadic Boolean algebras. In this volume Halmos collects his papers (unchanged) and adds a brief preface, a bibliography, and an index.

**7.4. Review by: The Editors.**

*Mathematical Reviews* MR0131961 **(24 #A1808)**.

The book consists of a collection of 10 papers of the author, together with a bibliography of 13 supplementary items, on algebraic logic.

**8. Lectures on Boolean algebras (1963), by Paul R Halmos.**

**8.1. From the Preface.**

In 1959 I lectured on Boolean algebras at the University of Chicago. A mimeographed version of the notes on which the lectures were based circulated for about two years; this volume contains those notes, corrected and revised. Most of the corrections were suggested by Peter Crawley. To judge by his detailed and precise suggestions, he must have read every word, checked every reference, and weighed every argument, and I am very grateful to him for his help.

**8.2. Review by: A D Wallace.**

*Science, New Series ***144** (3618) (1964), 531-532.

The literary style is a mature version of the author's earlier presentations and there is a trace of current Gallic hauteur as well as some Bourbakian pontification. ... But this is an entirely minor and subjective opinion, and the book is most highly recommended as a generally elegant and perceptive introduction to the basic facts concerning Boolean algebra available to the very well-trained senior and to the average second-year graduate student.

**8.3. Review by: R Sikorski.**

*Mathematical Reviews* MR0167440 **(29 #4713)**.

This is a very good introduction to the theory of Boolean algebras. It contains a discussion of all fundamental notions like subalgebras, homomorphisms, ideals, filters, complete algebras, √-complete algebras, etc. ... The book contains almost no references to the literature.

**8.4. Review by: R S Pierce.**

*The Journal of Symbolic Logic* **31** (2) (1966), 253-254.

The theory of Boolean algebras is one of the most attractive parts of mathematics. On the one hand, Boolean algebras arise naturally in such diverse fields as logic, measure theory, topology, and ring theory, so that the study of these objects is motivated by important applications. At the same time, the theory which has been developed constitutes one of the most elegant parts of modern algebra. Finally, the subject still poses many challenging questions, some of which have considerable importance. A graduate student who wishes to study Boolean algebras will find three excellent books to smooth his way ... For an introduction, the book by Halmos is probably the best of these monographs. It offers a quick route to the most attractive parts of the theory ... Like all of the books by Professor Halmos, Lectures on Boolean algebras is written with admirable style and clarity. This work is a welcome addition to the literature of mathematics.

JOC/EFR August 2016

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