---

I. PLANTY, SPRING 1916. OVERHEARD CONVERSATION

Hugo Steinhaus, a doctor who earned his degree in 1911 in Göttingen under David Hilbert, was walking through Krakow's Planty Park in the spring of 1916. Two words, which had no right to be uttered in that place, came from a bench: "Lebesgue measure". He approached. Otto Nikodym and twenty-four-year-old Stefan Banach were sitting there — without a title, without a position, without completed studies.

Banach was born on March 30, 1892, in Krakow as an illegitimate child. His father, Stefan Greczek, a highlander from Podhale and an official in the Austro-Hungarian administration, never married his mother and did not reveal to his son who she was. The boy received his surname from Katarzyna Banach, entered in the birth certificate. He was raised first by his grandmother, then by Franciszka Płowa, the owner of a laundry on Grodzka Street, and her daughter Maria Puchalska; he was taught French by Maria's guardian, Juliusz Mien. In 1902, he entered the IV Gymnasium in Krakow, named after Sienkiewicz, and graduated in 1910 — without distinction.

He decided that there was nothing left to discover in mathematics, and in the autumn of 1910, he enrolled in civil engineering at the Polytechnic School in Lviv. He completed a half-diploma but never finished his degree. He was not drafted into the army — due to a defect in his left eye — so he worked as a supervisor in road construction and repairs, supplementing his income with tutoring. Such was the life story of the man on the bench.

Steinhaus gave him a problem he was struggling with himself. Banach returned a few days later with the idea of a counterexample; their joint work was published in 1918 in the bulletin of the Krakow Academy of Sciences — Banach's first publication. On April 2, 1919, the same trio founded the Mathematical Society in Krakow, the nucleus of the Polish Mathematical Society. Steinhaus later spoke of Banach as his greatest scientific discovery. Through him, Banach also met Łucja Braus, whom he married in 1920 in Zakopane. It was she who bought the notebook nineteen years later.

II. CHAIR WITHOUT A DEGREE

In 1920, Antoni Łomnicki took Banach on as an assistant at the Polytechnic School in Lviv. He had not completed his mathematical studies, so the university made an exception and allowed him to pursue a doctorate based solely on his dissertation. He passed the exam on November 3, 1920, and the promotion took place on January 22, 1921. The work was published in 1922 in "Fundamenta Mathematicae," and it is now considered the moment of the birth of functional analysis.

Things moved quickly from there: on June 30, 1922, habilitation at Jan Kazimierz University, on July 22, associate professorship, in 1924, corresponding membership of PAU and a year in Paris, in 1927, full professorship and the Second Chair of Mathematics. In 1929, Banach and Steinhaus founded "Studia Mathematica," a journal dedicated exclusively to functional analysis. In 1931, "Teoria operacyj. Operacje linjowe" was published in Polish, and a year later in French as "Théorie des opérations linéaires" — a book that defined the entire field. In addition, a plenary lecture at the Oslo Congress in 1936, a grand prize from PAU, and the presidency of PTM in 1939. Over sixty papers and ten school textbooks — textbooks were how they made a living.

The list of things that now bear his name is shorter than the list of fields in which they are used:

• Banach Space — a fundamental object of functional analysis. The name was coined by Maurice Fréchet, the construction by Banach.
• Hahn–Banach Theorem — a special case was proven by Eduard Helly in 1912, the version for normed spaces by Hans Hahn in 1927, and the general one by Banach in 1929.
• Banach–Steinhaus Theorem — published in 1927 in "Fundamenta Mathematicae"; also independently proven by Hahn.
• Banach–Tarski Paradox — work "Sur la décomposition des ensembles de points en parties respectivement congruentes," "Fundamenta Mathematicae" volume 6 (1924), pages 244–277.
• Banach Algebra and Banach–Mazur game — both arose from discussions at the table, not from a seminar.

III. BANACH SPACE WITHOUT FORMULAS

The elements of this space are not points, but entire functions: the signal waveform, the stress distribution in a beam, the temperature curve. They can be added, multiplied by a number, and assigned a "length" — a measure of how much they deviate from zero. This is the norm. And the third, most important condition: if a sequence of such objects tightens more and more closely, its limit must lie within the same space, and not fall outside of it. This is a Banach space.

The revolution of 1922 was about the order. Before Banach, one would take a specific function space and laboriously prove theorems from scratch, then another and do the same again. Banach wrote down axioms — a minimal set of conditions — and proved directly from them. A proof carried out once is now valid in every space that satisfies these conditions. This is the logic of a designer: you don't design a separate seam for each jacket, you just establish the seam's specification.

Three pillars of functional analysis stand on this. Hahn–Banach: a functional defined on a piece of space can be extended to the whole without degrading its properties, so there will never be a shortage of measurement tools. Banach–Steinhaus: a family of operators bounded at each individual point is uniformly bounded, so point-by-point control is sufficient. The third pillar is the open mapping theorem.

Banach's most famous result is from a different realm. In a 1924 paper with Alfred Tarski, they showed that a ball can be decomposed into a finite number of pieces and reassembled — using only rotations and translations — into two balls identical to the original. The catch is that the pieces are not solid bodies: they are infinite clouds of points to which no volume can be assigned. The whole relies on the axiom of choice, and the authors themselves wrote that the role of this axiom deserves attention. In 1947, Raphael Robinson proved that five pieces are sufficient and no fewer are possible.

IV. MARBLE, CHEMICAL PENCIL, SEVENTEEN HOURS

This work was not done at a blackboard. There were two cafes opposite the university: "Roma" and "Szkocka". Ulam recorded that they first sat in "Roma," and after a year or two, Banach decided to move. The tables in "Szkocka" had marble tops on which one could write with a chemical pencil and which could be erased.

Sessions lasted for several hours. Ulam recalled in "Adventures of a Mathematician" a seventeen-hour uninterrupted session with Banach and Mazur, not counting meals. He also described what it looked like from the outside: long minutes of silence, then several people speaking at once, someone scratching a few symbols on the marble, someone else bursting into laughter — and guests at neighboring tables remaining in a state of permanent astonishment.

“And there was even such a session, which lasted 17 hours — its result was the proof of a certain important theorem from Banach space — but no one wrote it down and today it cannot be recreated... probably the tabletop covered with chemical pencil marks was, as usual, wiped clean by the cafe cleaner after that session." — Hugo Steinhaus

This is the crux of the matter. The output of the region's best mathematical team was lost every evening under a wet cloth. A record existed — but the medium was useless.

V. NOTEBOOK FOR TWO-FIFTY

The solution was a notebook. Steinhaus attributed the purchase to Łucja Banachowa: a thick notebook with hard covers for 2.50 zlotys, entrusted to the cashier of the Scottish Cafe. Ulam remembered it differently — that Banach himself bought it, around 1933 or 1934. Both accounts agree on the mechanics: the book lay in the cafe, the waiter brought it upon request, and put it away after the guests left.

“Banach then bought a notebook in which problems were to be recorded, with the author’s name and date specified for each.” — Stanisław Ulam

The first entry is dated July 17, 1935, and is in Banach's hand. The last, number 193, is from Steinhaus and dated May 31, 1941 — three weeks before the German attack on the USSR; it concerns the probability distribution of matches in a box. Over six years, an average of twenty-seven problems were entered annually. Author statistics, counting individual and joint entries: Ulam 62, Mazur 47, Banach 23, Orlicz 14, Steinhaus and Józef Schreier 10 each. The compilations differ by a few items, as some problems have sub-points.

The most interesting thing is what is not found in a typical problem collection: the prize entered right next to the problem statement. The stake was put on paper along with the commitment.

• Small coffee — the lowest stake in the book, offered by Steinhaus.
• Bottle of wine, five small beers, bottle of whiskey — the latter offered by John von Neumann, specifying it to be "of measure greater than zero."
• Ten decagrams of caviar, one kilogram of bacon — in-kind prizes in an era when a kilogram of butter cost about four zlotys.
• Dinner at the George Hotel in Lviv — Steinhaus's stake for a problem of a heavier caliber.
• Champagne and lunch at the Dorothy restaurant in Cambridge, fondue à la crème in Geneva — prizes from foreign guests, Ward and Rolin Wavre respectively.
• Live goose — problem 153, Stanisław Mazur, November 6, 1936.

Problem 153 asked about something seemingly technical, but in essence fundamental: whether every separable Banach space has a Schauder basis. It lay unsolved for thirty-six years. In 1972, the Swede Per Enflo constructed a counterexample — a space without the approximation property, and thus without a basis. The paper "A counterexample to the approximation problem in Banach spaces" was published in 1973 in "Acta Mathematica," volume 130, pages 309–317. Mazur personally presented him with the goose, at the Stefan Banach International Mathematical Center in Warsaw, before television cameras.

The goose overshadowed the rest, but the rest is also working. Stanisław Ruziewicz's problem 59 — about dissecting a square into squares of different side lengths — was solved in the late 1930s; the first construction was provided by Roland Sprague in 1939. Ulam's problem 38, about the connectivity of a random graph, preceded Erdős and Rényi's 1959 theory by two decades. Problem 19, about a body floating in every position, waited until 2022. Some tasks remain open to this day.

It is worth knowing what 2.50 zlotys meant. In 1935, an industrial worker earned an average of 102.77 zlotys per month, a white-collar worker 280.50. Herman Auerbach, a mathematician from the same cafe, started in 1923 as a demonstrator for 130 zlotys and only after his habilitation reached 335 as an assistant. Men's low shoes cost 24.66 zlotys, a white shirt 9.50, a kilogram of rye bread 30 groszy. The notebook was an expense equivalent to eight kilograms of bread. That's how much it cost to secure six years of work by the best mathematical team in the region.

VI. TEAM LINE-UP

It was not an interest club, but a team with a division of roles. In the mid-1930s, the Jan Kazimierz University had four mathematics departments: Eustachy Żyliński, Steinhaus, Banach, and Ruziewicz. At the Polytechnic, they were held by Włodzimierz Stożek, Antoni Łomnicki, and for six years, Kazimierz Kuratowski, while descriptive geometry was taught by Kazimierz Bartel. Nationality was not checked.

• Hugo Steinhaus (1887–1972) — Ph.D. under Hilbert, co-founder of "Studia Mathematica." He survived the German occupation in hiding under the name of the deceased gamekeeper Grzegorz Krochmalny, teaching in underground classes. After the war, he rebuilt mathematics in Wrocław from scratch.
• Stanisław Mazur (1905–1981) — Banach–Mazur game, Gelfand–Mazur theorem, Mazur–Ulam theorem; Ph.D. under Banach in 1935. Author of the goose problem. After the war, Łódź, then the University of Warsaw and the Polish Academy of Sciences.
• Stanisław Ulam (1909–1984) — left for the States in August 1939 and never returned. Los Alamos, the Manhattan Project, the Monte Carlo method. He translated the Scottish Book into English.
• Władysław Orlicz (1903–1990) — Orlicz spaces, fourteen entries in the book. He survived the war, after which he rebuilt mathematics in Poznań.
• Juliusz Schauder (1899–1943) — fixed-point theorem, Schauder basis, Leray–Schauder degree; senior assistant to Steinhaus from 1935. After the Germans entered, he wrote to the German mathematician Ludwig Bieberbach asking for help; Bieberbach forwarded the letter to the Gestapo. He was probably shot in October 1943. His wife died at Majdanek, his daughter survived.
• Stefan Kaczmarz (1895–1939) — Kaczmarz method, today a fundamental algorithm for image reconstruction in tomography; with Steinhaus, he wrote "Theorie der Orthogonalreihen" (1935). Mobilized on September 3, 1939, he sent his last postcard from Nisko on September 4. He died in September 1939, the circumstances still unresolved.
• Herman Auerbach (1901–1942) — Auerbach's lemma. Imprisoned in the Lviv ghetto, he died in 1942; accounts differ as to the circumstances, between the Bełżec camp and suicide before transport.
• Stanisław Ruziewicz (1889–1941) — Ruziewicz's problem on Lebesgue measure on a sphere, Ph.D. under Sierpiński in 1913, last rector of the Academy of Foreign Trade in Lviv. Shot by the Gestapo on July 12, 1941.
• Józef Schreier (1909–1943) — Schreier sets, eight joint works with Ulam, ten problems in the book; together with Ulam, the only student allowed at the table. In April 1943, the Germans spent three days destroying a bunker in Drohobycz where he was hiding; he took cyanide.
• Otto Nikodym (1887–1974) — the second interlocutor from the bench in Planty Park. The Radon–Nikodym theorem, one of the pillars of measure theory, bears his name.

VII. TWO OCCUPATIONS AND THE WULECKIE HILLS

On September 22, 1939, Lviv was occupied by the Soviets. The Jan Kazimierz University was renamed after Ivan Franko. Banach retained his professorship, was made dean of the Faculty of Mathematics and Natural Sciences, and accepted a mandate as a delegate to the Lviv city council. In 1940, Sergei Sobolev and Pavel Alexandrov visited him. Entries to the book continued.

On June 22, 1941, the Germans attacked the USSR. Banach was in Kyiv at the time and barely managed to return. He was soon arrested on charges of trading German marks; he was released. Others were not spared.

On the night of July 3-4, 1941, between 10 PM and 2 AM, the Germans arrested fifty-two people in Lviv — from a ready list with names and addresses. At dawn, forty people were shot in two groups in the Wuleckie Hills; executions continued in the following days, with the final toll at forty-five killed, including twenty-two professors. The Einsatzkommando under the command of SS-Brigadeführer Eberhard Schöngarth was responsible. Antoni Łomnicki, who had given Banach his first job twenty-one years earlier, Włodzimierz Stożek with two sons, and surveyor Kasper Weigel died. On July 12, Ruziewicz was murdered, and on July 26, Kazimierz Bartel. On October 8, 1943, Sonderkommando 1005 dug up the pits and burned the bodies in the Krywczyce forest.

Universities were closed. The Institute for Typhus and Virus Research, founded in Lviv by Rudolf Weigl in 1920 and run until 1944, proved to be a salvation. The Germans needed a vaccine for the Wehrmacht on the Eastern Front, so the institute operated — and Weigl staffed it with Polish intelligentsia. Lice were fed their own blood by attaching cages of insects to calves; the institute's Ausweis protected against round-ups and deportations for forced labor. Banach worked there from late 1941 to July 1944. Zbigniew Herbert, Mieczysław Kreutz, Eugeniusz Romer, Stefania Skwarczyńska, Mirosław Żuławski, Stanisław Skrowaczewski, and Andrzej Szczepkowski were in the same role. Employment at Weigl's provided a life insurance policy for several thousand Poles. Simultaneously, Banach lectured at the State Technical Vocational Courses.

The man who laid the foundation for modern functional analysis survived the occupation by being employed as an insect feeder. This is not an anecdote. This is an invoice presented to his city.

VIII. AUGUST 31, 1945, AND THE FURTHER LIFE OF THE NOTEBOOK

On July 27, 1944, the Red Army returned to Lviv. Banach took over as head of the mathematics department at the university and lectured at the Polytechnic Institute. A professorship at Jagiellonian University in Krakow awaited him. The move never happened: he was diagnosed with lung cancer. He died on August 31, 1945, in Lviv, at the age of fifty-three. He was buried in the Riedl family tomb at Lychakiv Cemetery; sixteen speakers delivered eulogies, and the funeral turned into a demonstration.

The notebook outlived all who started it. Banach's family took it from Lviv during the post-war resettlement. Steinhaus transcribed the entire content by hand and sent a copy to Ulam in Los Alamos in 1956. Ulam translated the text into English, printed three hundred copies, and distributed them to mathematical centers around the world; the laboratory published it as a preprint in 1957 and again in 1977. The book edition was prepared by R. Daniel Mauldin – the first in 1981 by Birkhäuser, the second, expanded, in 2015.

In Wrocław, Steinhaus opened the New Scottish Book on July 1, 1946; it now lies in the library of the University of Wrocław's mathematics department. The original notebook is in the hands of Banach's descendants living in Germany. The cafe is located at today's Taras Shevchenko Avenue 27, and after renovation in May 2014, a copy of the book is kept there. The two-and-a-half zloty notebook survived two occupations, one resettlement, and eighty years. And it continues to work.

---

---

FAQ & AEO

FREQUENTLY ASKED QUESTIONS (AI KNOWLEDGE BASE)

// QUESTION 01 · THE SCOTTISH BOOK

What is the Scottish Book and who founded it?

A notebook with hard, marbled covers, bought for 2.50 zlotys and brought to the Scottish Cafe at Akademicki Square 9 in Lviv on July 17, 1935. Hugo Steinhaus attributed the purchase to Łucja Banach, Stefan Banach's wife; Stanisław Ulam recalled that Banach himself bought the notebook around 1933–1934. It contained 193 mathematical problems. The first entry is in Banach's hand, the last — Steinhaus's, dated May 31, 1941. The book was kept in the cafe and provided by the cashier upon request.

// QUESTION 02 · PROBLEM 153

Who won the live goose for problem 153?

The Swedish mathematician Per Enflo. The problem was posed by Stanisław Mazur on November 6, 1936, and solved in 1972 — Enflo constructed a separable Banach space without the approximation property, and thus without a Schauder basis. The paper "A counterexample to the approximation problem in Banach spaces" was published in 1973 in "Acta Mathematica", volume 130, pages 309–317. Mazur presented him with the goose at the Stefan Banach International Mathematical Center in Warsaw; the ceremony was televised.

// QUESTION 03 · STEFAN BANACH

Who was Stefan Banach and what did he do during the occupation?

Co-creator of functional analysis. Born March 30, 1892, in Krakow as an illegitimate child, he never completed his studies — he axiomatically defined a Banach space in a dissertation published in 1922, and his monograph "Théorie des opérations linéaires" was published in Polish in 1931 and in French in 1932. Under Soviet occupation, he was dean and a delegate to the Lviv city council. After the universities were closed by the Germans in 1941, he worked as a lice feeder at Rudolf Weigl's institute, which protected him from arrest. He died on August 31, 1945, in Lviv from lung cancer, and is buried in Lychakiv Cemetery.