## Reviews of books by Roman Sikorski

**1. Real functions (Polish) (1958), by Roman Sikorski.**

**1.1. Review by: Stanislaw Ulam.**

*Bull. Amer. Math. Soc*.

**65**(1959), 305-306.

The introduction to this Monograph contains an account of the ideas and theorems of set theory and topology, necessary for the understanding of the subject treated in the book, which is divided into three parts. In the first part, the idea of continuity of a function is studied extensively. The second part develops Lebesgue's integration, together with a detailed treatment of abstract measure theory and the Stieltjes integral. ... The third part of the book deals with applications of the theory of Lebesgue integration to orthogonal series, and to Fourier series and the Fourier integral. The treatment of all this material is outstanding by its great clarity and in showing how the deeper results of set theory and the abstract theory of measure find applications in functional analysis in general, and the theory of orthogonal series. The theorems are presented in as general a form as possible without destroying the simplicity of their formulation.

**1.2. Review by: Z A Melzak.**

*Mathematical Reviews* MR0091312 **(19,945a).**

This book contains the first two of the three parts of a treatise on the theory of functions of a real variable. Of its eleven chapters, the first two are an introductory exposition of sets, functions, classes of sets, and metric spaces; the next three chapters deal with continuity and convergence, and the remaining six are devoted to measure, integration, and differentiation. ... As for the general value of the book - unlike certain venerable (though outdated) Englishmen, the author has no desire to say everything, succeeding in it better, and unlike certain nameless Frenchmen, he confers mathematical maturity instead of demanding it. In spite of the modest claims in the preface, the completed work will represent an outstanding treatise and text.

**2. Real functions Vol. 2 (Polish) (1959), by Roman Sikorski.**

**2.1. Review by: Stanislaw Ulam.**

*Bull. Amer. Math. Soc.*

**67**(1959), 172.

The second volume of Sikorski's *Real functions* deals with function spaces, orthogonal series with special treatment of Fourier series and Fourier integrals. ... This volume continues the excellent treatment characterizing the first part - in its modern point of view and the concise and elegant development of recent results.

**2.2. Review by: Z A Melzak.**

*Mathematical Reviews* MR0105468 **(21 #4209).**

This completes the author's treatise on real variables ... There are four chapters: Function spaces; Hilbert space; Fourier series; Fourier integrals. ... Although this volume seems to be somewhat inferior to the first one, the two form an excellent treatise and the Polish students and mathematicians are to be congratulated.

**3. Boolean Algebras (1960), by Roman Sikorski.**

**3.1. Review by: Bernhard H Neumann.**

*The Mathematical Gazette*

**45**(354) (1961), 365-366.

The appearance of this book in the well-known series "Ergebnisse der Mathematik" is timely. The author's point of view is topological rather than algebraic (though algebraic topics are by no means neglected); the justification for this is to be found in the second chapter, which deals with infinitary operations and is the most weighty and original part of the book and occupies half of its bulk

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

*Mathematical Reviews*** **MR0126393 **(23 #A3689).**

There has long been a need for a book on the mathematical theory of Boolean algebras at the graduate or research level. The author's book fills this need in spectacular fashion. In two chapters and an appendix of 160 pages almost all of the important theorems on Boolean algebras are presented. What is more remarkable, most of these results are proved in the book.

**4. Boolean Algebras (Second Edition) (1964), by Roman Sikorski.**

**4.1. Review by: Donald Monk.**

*The Journal of Symbolic Logic*

**31**(2) (1966), 251-253.

... the second edition is much more complete than the first. The second edition gives, in fact, a comprehensive survey of the most well-developed aspect of the theory of Boolean algebras, namely the set-theoretical and topological one. Little mention is made of the ring-theoretic aspect, and no mention is made of applications to circuit theory. ... Innumerable important detailed and specialized theorems and examples are found throughout. The book, which has already become a classic in its field with the first edition, should serve as the main reference and inspiration for anyone interested in the subject.

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

*Mathematical Reviews*** **MR0177920 **(31 #2178).**

The growth of the theory of Boolean algebras during the past few years is clearly seen by comparing the sizes of the first and second editions of the author's monograph. The first edition contained 176 pages; the new edition, written only four years later, has 237 pages. This development is paralleled by an increase in the literature on Boolean algebras. There are twelve pages of bibliography in the author's original work, and nineteen in the revision. ... The new edition of the author's distinguished book brings the story of Boolean algebras pretty much up to date. In doing this, it retains all of the virtues of the original work. This monograph is amazingly complete. Everything that the research worker needs to know about Boolean algebras can be found between its covers. Still more remarkable is the fact that the book could serve very well to introduce a serious student to a wide range of topics in set theory, topology, measure theory, logic, and, of course, the theory of Boolean algebras.

**5. The Mathematics of Metamathematics (1963), by Helena Rasiowa and Roman Sikorski.**

**5.1. Review by: Donald Monk.**

*The Journal of Symbolic Logic*

**32**(2) (1967), 274-275.

This book is intended as an introductory account of general algebraic, lattice-theoretical, set-theoretical, and topological methods in metamathematics. Although it can serve as a textbook for mathematically trained beginners, it should be mentioned at the outset that it is not a conventional survey; areas in metamathematics that have not yet been treated by these methods are not discussed at all. This means first of all that no Gödel-type incompleteness or undecidability results are given. In the realm of the methods with which the book deals the authors have restricted themselves to their own research, or closely related work; for this reason a great deal of modern model theory is omitted from the book. Within these limitations, however, the authors have given quite a broad survey of contemporary metamathematics, and the topics treated are done so with elegance. Moreover, several new ideas of the authors are clearly exposed in the book, whereas formerly they have been buried in the large number of specialized publications of the two authors in this area. ... In summary, the book can serve as a good introduction to infinitistic methods in metamathematics, and as a reference for some topics that are rarely treated outside of research papers.

**5.2. Review by: G Kreisel.**

*Mathematical Reviews*** **MR0163850 **(29 #1149).**

This very agreeable book gives a systematic and leisurely account of the authors' researches in elementary metamathematics; elementary in the sense that it is almost wholly concerned with various predicate logics of first order. ... Most of the specific results in the book seem to have been published previously; the authors do not indicate what is new over their previous articles. But the coherent and smooth presentation of the whole body of material is, of course, more effective than a series of research papers. The other reason why this book is so engaging is the lucid and frank formulation of the authors' view on the relative role of mathematics and philosophy in logical research, a view which is widely held, but rarely put so clearly. Roughly speaking, they consider it a miracle, evidently beyond rational explanation, that general and "vague'' philosophical conceptions should lead to such rich formal structures as intuitionistic formal systems.

**6. Differential and integral calculus: Functions of several variables (Polish) (1967), by Roman Sikorski.**

**6.1. Review by: S Mandelbrojt.**

*Mathematical Reviews*MR0213483

**(35 #4346).**

This volume is one of the few contemporary treatises of the differential and integral calculus in the theory of functions of several variables that is designed in a very modern way, in the subject matter as well as the method and language used. The author willingly uses the language of linear algebra; symbolism - matrices, vectors - allows the author to express formulas in the space of several dimensions in a manner analogous to that of the standard differential and integral calculus.

**7. Advanced Calculus. Functions of Several Variables (1969), by Roman Sikorski.**

**7.1. From the Preface.**

This is a translation of the second Polish edition of *Differential and integral calculus: Functions of several variables* (Polish) (1967).

**7.2. Review by: R C Buck.**

*Amer. Math. Monthly* **79** (8) (1972), 921-923.

The presentation of analysis in colleges is tied to a number of factors - the career goals of students, their prior preparation and mathematical aptitudes, and the need to articulate with other courses in mathematics and science. In spite of its title, the present book does not seem to have a place in such a programme. It is the result of a desire by the author for a treatment in which everything is done "right" the first time. While the objective may be attractive, its accomplishment is far from easy; in the present attempt, the viewpoint and notation quickly become uncompromising and the atmosphere austere and oppressive. ... This book seems to reflect what I feel to be an unfortunate belief among some teachers of mathematics that the proper way to generate research mathematicians is by presenting students with the most general and abstract version of a topic. When this is done, the basic ideas of the subject are apt to get lost amid a welter of notation and structure. There is no high points or climaxes, no broad over-view, no explanation of a simple illustration before meeting the full treatment. Such a process may be a logical way to organise the subject, but I feel it is a very poor to teach students to be creative, or to understand the origins and significance of the subject. The best mathematics is seldom turgid.

**8. Introduction to differential geometry (Polish) (1972), by Roman Sikorski.**

**8.1. Review by: V Komkov.**

*Mathematical Reviews*MR0467544

**(57 #7400).**

This text is an outgrowth of lectures at the undergraduate level at Warsaw University. Since the students had no preparation in algebraic topology, particularly in the theory of fibre bundles, the exposition follows the use of the concepts of the classical tensor calculus, but with a unique modern flavor. The approach is basically algebraic. ... This is an outstanding expository text, available in the form of lecture notes since 1969.

**9. Theory of distributions. The sequential approach (1973), by Piotr Antosik, Jan Mikusinski and Roman Sikorski.**

**9.1. Review by: Z Zielezny.**

*Mathematical Reviews*

**MR0365130**

**(51 #1383).**

This book is devoted to the theory of distributions defined as equivalence classes of sequences of continuous functions.

JOC/EFR February 2017

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