Pearls from a lost city. The Lvov School of Mathematics. Translated from the Polish by Daniel Davies.

*(English)*Zbl 1303.01001Between World War I and World War II, the (then) Polish city of Lwów hosted one of the world centres of mathematics. It can be considered as one of the birthplaces of functional analysis, and a big part of the theory of normed spaces was developed there. Luminaries like Stefan Banach, Hugo Steinhaus, Kazimierz Kuratowski or Juliusz Schauder held positions in Lwów, and among the mathematicians of the next generation for example Mark Kac and Stanisław Ulam belonged to the Lwów school.

Roman Duda’s book tells the story of this school from its beginning (essentially dating back to Banach’s dissertation in 1920) until its destruction by the Nazis, describing its protagonists (Chapter 2) and its main achievements (Chapter 3); in particular, the author highlights Steinhaus’s merits for the mathematical foundations of probability theory. Most importantly, he also discusses social aspects in the broad sense, e.g., the development of the Polish Mathematical Society, working habits (the Scottish Café and the Scottish Book), the lack of proper positions for researchers (Schauder had to make a living as a school teacher for many years), etc. Among these social aspects one should also mention the prehistory of the Lwów school (Chapter 1), e.g., Janiszewski’s programme to concentrate mathematical research in Poland on relatively few areas in which Polish mathematicians excelled, like set theory and topology. Later, functional analysis became such a field as well.

Chapter 4, entitled “Oblivion”, deals with the end of the Lwów school after the first Soviet occupation (1939–41), the German occupation (1941–44), during which the Polish intelligentsia was virtually exterminated, and the second Soviet occupation and the ensuing expulsion of the Polish population in 1945. Further, there are appendices listing mathematicians associated to Lwów and their works.

The style of the book is rather matter-of-fact; various anecdotes found in other sources are not reproduced here, and somewhat unfavourable traits are omitted (for example Banach’s drinking habits). Understandably, the book’s focus is on the achievements of the Lwów school (of which there are many) rather than its shortcomings; for example, although it is pointed out that Banach only came in second in proving the Hahn-Banach extension theorem for normed spaces, there is no mention of his formal “Reconnaissance du droit de l’auteur” published in [S. Banach, Stud. Math. 2, 248 (1930; JFM 56.0368.06)].

Like every book, this one is not entirely free of (mostly inessential) errors either: Per Enflo’s name is misspelt, the Kadets-Anderson theorem (page 107) assumes separability (the general case is due to Torunczyk), the Lindenstrauss-Tzafriri volumes (page 119) were not updated for the Classics in Mathematics reprint (much to the authors’ chagrin), and the formulation of the inverse mapping theorem on page 106 contains the right words, but in the wrong order.

Many journal articles have been devoted to various aspects of mathematics in Lwów or to biographies of Lwów mathematicians, but Duda’s book is the first comprehensive exposition. It is a must-read for everyone interested in the history of functional analysis or of mathematics in Poland, where the original Polish edition from 2007 [Lwowska Szkoła Matematyczna (Polish). Wrocław: Wydawnictwo Uniwersytetu Wrocławskiego (2007; Zbl 1296.01004)] has been highly successful. There is good reason to assume that the English version will be likewise successful.

Roman Duda’s book tells the story of this school from its beginning (essentially dating back to Banach’s dissertation in 1920) until its destruction by the Nazis, describing its protagonists (Chapter 2) and its main achievements (Chapter 3); in particular, the author highlights Steinhaus’s merits for the mathematical foundations of probability theory. Most importantly, he also discusses social aspects in the broad sense, e.g., the development of the Polish Mathematical Society, working habits (the Scottish Café and the Scottish Book), the lack of proper positions for researchers (Schauder had to make a living as a school teacher for many years), etc. Among these social aspects one should also mention the prehistory of the Lwów school (Chapter 1), e.g., Janiszewski’s programme to concentrate mathematical research in Poland on relatively few areas in which Polish mathematicians excelled, like set theory and topology. Later, functional analysis became such a field as well.

Chapter 4, entitled “Oblivion”, deals with the end of the Lwów school after the first Soviet occupation (1939–41), the German occupation (1941–44), during which the Polish intelligentsia was virtually exterminated, and the second Soviet occupation and the ensuing expulsion of the Polish population in 1945. Further, there are appendices listing mathematicians associated to Lwów and their works.

The style of the book is rather matter-of-fact; various anecdotes found in other sources are not reproduced here, and somewhat unfavourable traits are omitted (for example Banach’s drinking habits). Understandably, the book’s focus is on the achievements of the Lwów school (of which there are many) rather than its shortcomings; for example, although it is pointed out that Banach only came in second in proving the Hahn-Banach extension theorem for normed spaces, there is no mention of his formal “Reconnaissance du droit de l’auteur” published in [S. Banach, Stud. Math. 2, 248 (1930; JFM 56.0368.06)].

Like every book, this one is not entirely free of (mostly inessential) errors either: Per Enflo’s name is misspelt, the Kadets-Anderson theorem (page 107) assumes separability (the general case is due to Torunczyk), the Lindenstrauss-Tzafriri volumes (page 119) were not updated for the Classics in Mathematics reprint (much to the authors’ chagrin), and the formulation of the inverse mapping theorem on page 106 contains the right words, but in the wrong order.

Many journal articles have been devoted to various aspects of mathematics in Lwów or to biographies of Lwów mathematicians, but Duda’s book is the first comprehensive exposition. It is a must-read for everyone interested in the history of functional analysis or of mathematics in Poland, where the original Polish edition from 2007 [Lwowska Szkoła Matematyczna (Polish). Wrocław: Wydawnictwo Uniwersytetu Wrocławskiego (2007; Zbl 1296.01004)] has been highly successful. There is good reason to assume that the English version will be likewise successful.

Reviewer: Dirk Werner (Berlin)