This book is a slightly expanded English translation of a Polish text of 2007, detailing the achievements of the Lvov mathematical school between the two world wars. As such, it brings to a wider audience the details of a group of mathematicians who are (with some exceptions) not so well known: much has been written on the corresponding group of mathematical researchers in Warsaw, so the author's aim was to plug what has hitherto been a Lvov-shaped gap in the literature. Since Polish Lvov became Soviet Ukrainian Lviv in 1946, the details of the Lvov school of mathematics have apparently been rather harder to piece together than those of its Varsovian counterpart.

The contents of the book are distributed across seven broadly defined parts, most of these then being further subdivided into chapters dealing with particular topics. The first five of these parts describe the school itself. Part I, for example, provides us with some background to the development of the Lvov mathematical school in the twentieth century by briefly outlining the details of the foundation of Lvov University (1661), the creation of its mathematics department (1744) and its establishment as a centre for mathematical research in the late nineteenth century. The later chapters of Part I bring the story up to about 1920, at which point we pass to Part II and the so-called ‘Golden Age’ of the Lvov mathematical school.

Part II focuses on the individual members of the school and the way in which they formed a community of researchers. We learn, for example, about the nature of the mathematical studies undertaken by research students in Lvov, and the ways in which mathematical ideas were communicated: through the foundation of the journal Studia Mathematica (now published in Warsaw), or via talks at the Lvov branch of what became the Polish Mathematical Society, for instance. We also find details here of the mathematical social life that centred upon Lvov's Scottish Café, and of the problems that were recorded in a notebook kept there (the so-called ‘Scottish Book’). The discussion of the ‘Golden Age’ continues in Part III, where the focus shifts to a treatment of the mathematical achievements of the Lvov school. Many of these were in functional analysis, but successive chapters outline the contributions of Lvov mathematicians to other areas, including probability, measure theory and operator theory.

The general tone of the earlier parts of the book is one of pride in the advances made by mathematicians based in Lvov during the years in question, but Part IV (‘Oblivion’) becomes rather more sombre, as we learn of the ultimate fate of the Lvov school, and of its individual members. Lvov was seized by the Red Army in 1939, after which a process of ‘Ukrainianization’ began; education was restructured along Soviet lines, and the Polish university was replaced by a Ukrainian one, which nevertheless employed many of the staff of its predecessor. Between 1941 and 1944 Lvov found itself under German occupation, during which period many university professors were summarily executed. The author's period of interest ends with the re-establishment of Soviet control over Lvov in 1945, and the expulsion of much of the remaining Polish population. Although mathematical research was later carried on in the renamed Lviv, the author takes 1945 as the end of the Lvov mathematical school.

In Part V, the author attempts to assess the historical significance of the Lvov mathematical school, not only by drawing conclusions from the material in the earlier parts of the book, but also by presenting extracts from the evaluations of the work of Lvov mathematicians that have been written by independent commentators over the decades. In looking for the impact of the Lvov school on world mathematics, the author reaches the conclusion that this was greatest in connection with the school's main specialism: functional analysis.

Part VI consists of an alphabetical list of brief biographies of some of the main figures to have appeared in the book, whilst Part VII is an extremely detailed bibliographical section, featuring, alongside the main bibliography of the book, a list of works by Lvov mathematicians, and one of personal recollections by members of the Lvov school and their collaborators.

The translation is mostly very readable, although it is a little clumsy in places: the original Polish sentence structure appears to have been followed too closely in such instances, resulting in slightly awkward English phrases. The occasional misuse of prepositions probably has a similar explanation. Another feature that grates slightly is the author's occasional fervent insistence upon asserting Polish priority in a given area of mathematics (even when, as far as the reviewer is aware, such priority was never in doubt). A further minor gripe is that various quantities of money (for example, professors' salaries) are quoted only in zloty; it would have been useful to have had more context for the Polish currency of the times – perhaps the price of a loaf of bread?

The criticisms made in the preceding paragraph are, however, only slight blemishes on an otherwise excellent book. There is much fascinating information here, some of it presented several times in different forms: in the prose of the earlier chapters, in the chronological overview that comes in Chapter 27, within the brief biographies of individual mathematicians and so on. This feature means that the book will be extremely valuable as an easily navigable reference source on Polish mathematics generally, and on that in interwar Lvov in particular. The author should be thanked for bringing this material to wider attention.