The focus of this book is on the mathematical activity and the mathematicians in the city of Lvov during the period 1920 to 1940, especially at Lvov University. The author refers to this period in Lvov as “The Golden Age.” Useful historical background is given concerning Lvov University and mathematical developments in Lvov and in Poland in general. Three chapters are devoted to the period from 1940 through early 1946, when Lvov was occupied and ruled sequentially by the Soviet Union, Nazi Germany, and then again by the Soviets. Duda calls this period in Lvov “oblivion.” The book is subdivided into seven major parts, each part in turn subdivided into chapters or sections. These major parts are: Part 1. Background; Part II. The Golden Age: Individuals and Community; Part III. The Golden Age: Achievements; Part IV. Oblivion; Part V. Historical Significance; Part VI. List of Lvov Mathematicians; Part VII. Bibliographies. Part VI gives brief biographies of each of the mathematicians associated with Lvov, over fifty in all. Part VII has three sections of bibliographies: (A) Mathematical Work by Lvov Mathematicians; (B) Personal recollections, Surveys, and Historical Source Material; and (C) Other Mathematical Works Cited.

Lvov University was established in 1661. In 1920 it was renamed Jan Kazimierz University (abbreviated UJK). It was the main center for mathematical activity in Lvov. Also important was Lvov Polytechnic (PL). The list of eminent mathematicians working in Lvov during the Golden Age is long and impressive. The brightest light in this constellation of mathematical stars was Stefan Banach. Banach was the premier functional analyst in the world from 1922 to 1945. There were many other outstanding functional analysts in Lvov during this period, including Kaczmarz, Mazur, Orlicz, and Steinhaus. This extraordinary group arguably made Lvov the preeminient center for functional analysis in the world during that period.

Banach lived in Lvov from 1910 until his death in 1945. He earned his doctorate and obtained his habilitation at UJK and became a professor there in 1923. He is best known for his work on complete normed linear spaces, which he called “spaces of type (B),” and which became known as “Banach spaces.” The latter terminology was used by Steinhaus in 1929, and rapidly became the standard terminology. His name is attached to many of the most important and beautiful results in functional analysis, e.g., Banach’s fixed point theorem, the Hahn-Banach Theorem, and the Banach-Steinhaus Theorem.

There were many other areas of mathematics in which the mathematicians in Lvov during the Golden Age made important and exciting contributions. Seminal work in measure theoretic probability theory was done by Steinhaus and A. Lominicki, as early as 1920–1923. New and surprising measure-theoretic results were obtained by Banach and Tarski, who visited Lvov and had a working partnership with Banach. Substantial work was done by Juliusz Schauder in nonlinear functional analysis and the theory of partial differential equations. Stanislaw Mazur published seminal work on normed algebras (what later became known as Banach algebras) in 1938. In 1937, Stefan Kaczmarz published a method that provides approximate solutions to systems of linear algebraic equations, now known as Kaczmarz’s algorithm. Kazlmierz Kuratowski, who was a professor at PL during 1927–1933, published work on graph theory in 1930. The notion of a locally convex linear topological space was discovered by Mazur in the early 1930s, independently of the work of von Neumann and Kolmogorov elsewhere. The stories of this work and of some other mathematical work during the Golden Age in Lvov is given in Part III. Because the mathematics is treated in more detail in this part, a greater level of mathematical maturity is needed by the reader for Part III. Graduate level courses in real analysis and functional analysis will safely carry the reader through. For the rest of the book the background of an upper level undergraduate math major will suffice.

Duda writes that “The greatness of the Lvov School can be seen in the intellectual audacity of its founders, H. Steinhaus and S. Banach.” This is exemplified by their free and bold use of non-constructive methods which establish existence, such as the axiom of choice and Baire category theory. For example, the Baire category theorem was used by Banach and Mazurkiewicz in a series of papers to prove the existence of continuous functions “having various curious features.” Others in the Lvov school made use of these nonconstructive techniques, e.g., Auerbach and Kaczmarz.

The social life of the Lvov mathematicians was intimately connected with their mathematics. In no place is this more evident than at the Scottish Café. Ulam commented that a significant part of the mathematical discussions took place in cafés neighboring the university. Most important of these by far was the Scottish Café. Excited mathematical conversation, some long periods of silent thought, and vast amounts of coffee went into the producing of much new and exciting mathematics. At first it was written on the table tops, but in 1935 Banach’s wife, Lucja, provided a thick exercise book to be kept in the Scottish Café for the mathematicians to use to record their problems, conjectures, and comments. The book became known as “The Scottish Book.” It was available to any mathematician who asked for it at the café. Some of the problems had prizes for their solution, ranging from a cup of coffee to a live goose.

The Scottish Book was in use for about six years, from July 17, 1935 to May 31, 1941. Many renowned mathematicians who visited Lvov contributed to the Scottish Book, e.g., John von Neumann and S. Sobolev. Duda devotes chapter 10 to the Scottish Café mileu and the Scottish Book. Today the original Scottish Book is in the custody of the Banach family; a copy is in the Library of the Mathematical Institute of the Polish Academy of Sciences in Warsaw. Duda makes clear that in the “Golden Age” in Lvov doing mathematics was often mixed with humor, friendships, and competitions.

Of importance in spreading the word about mathematics done at Lvov to all of Europe and the world was the journal *Studia Mathematica*, which was based in Lvov and was published from 1929 until 1940. Banach and Steinhaus were the founders and first editors of this journal. Also helping to make Lvov known as a mathematical center was the monograph series *Monografie Matematyczne*. The first in the series was Banach’s very influential work *Théorie des Operations Lineaires*, 1932. One other monograph in the series was *Theorie der Orthogonalreihen*, by Kaczmarz and Steinhaus, 1936. Two others in the series were written by authors connected with Lvov: Kuratowski, *Topologie I*, 1933, and A. Zygmund, *Trigonometrical Series*, 1935.

On the eve of World War II, Lvov, with about 350,000 inhabitants, was a major academic and cultural center in Poland. In addition to UJK and PL, Lvov had several professional schools. Just over six years later Lvov was no longer a part of Poland, but was part of the Ukrainian republic of the U.S.S.R., with a population of barely 30,000. Most of the Poles had either been expelled from the Lvov region or had died. And the mathematical culture of Lvov was in “oblivion.”

On completing Duda’s book one should be convinced that Lvov during the period 1920–1940 was the setting for much extraordinary and even epoch making mathematics. This was set in a mathematical community which must have been exciting and stimulating to be part of. It is sad that the evil regimes of Nazi Germany and the Soviet Union destroyed it.

The book under review is well and carefully written. The translation from Polish into English is polished and lively. The only aspect in which I find the book needs improvement is that it has no subject index. The book does have an index of names. I highly recommend the book for all university libraries, and I recommend it to those interested in the history of mathematics. The general mathematical reader will find it an entertaining and informative story about mathematicians and a truly extraordinary mathematical community. For those interested in further reading about mathematicians and mathematics in Lvov and in Poland in the first half of the 20th century see the references listed below.

**References**

[1] Krzysztof Cielieski, “Lost Legends of Lvov 1: The Scottish Café,” *Math. Intelligencer*, **9** (1987), 36–37.

[2] Emila Jakimowica and Adam Miranowicz, *Stefan Banach: Remarkable Life, Brilliant Mathematics*, Gdansk Univ. Press, Gdansk, 2011.

[3] Roman Kaluza, *Through a Reporter’s Eyes. The Life of Stefan Banach*, Birkhäuser, Boston, 1996.

[4] Kazimierz Kuratowski, *A Half Century of Polish Mathematics: Remembrances and Reflections*, Pergamon Press, Oxford, 1980.

Henry Heatherly is Emeritus Professor of Mathematics at the University of Louisiana, Lafayette. His current research interests are rings and semigroups. Each Spring semester he teaches a course in the history of mathematics.