**Stanisław Mazur**attended the Gymnasium in Lvov, graduating in 1923. From 1923 to 1926 he studied at the Jan Kazimierz University, Lvov and in Paris. In [1] an incident from this time is recalled by Andrzej Turowicz. Mazur wrote his first paper while still an undergraduate and he submitted it to Steinhaus. The paper was to be read by Mazur at a meeting of the Lvov Scientific Society but only hours before the meeting Steinhaus summoned Mazur to tell him that he had handed him four blank sheets of paper. Students could only afford cheap quality yellow paper and Mazur had diluted his ink with water to make it last longer. Steinhaus said to Mazur:-

There is no record of whether Mazur took Steinhaus's advice, but he certainly devoted himself to mathematical pursuits. In the years 1926-1935 he was an assistant to the chair of mathematical analysis at the Jan Kazimierz University. He became a student of Banach's who taught there.Well, Mr Mazur, perhaps there is something written here after all. But if you intend to devote your life to scientific pursuits, why don't you first supply yourself with white paper and black ink.

His doctorate, under Banach's supervision, was awarded in 1935. Mazur was a close collaborator with Banach at Lvov and became a member of the Lvov School of Mathematics, a group of about a dozen mathematicians working in functional analysis, real functions and probability theory. He wrote several papers in collaboration with Banach during the 1930s and, having a better knowledge than Banach of German, he polished the language used in the joint papers which they wrote in German.

The collaboration between Mazur and Banach in Lvov was important for both men. There is no doubt that of all Banach's colleagues in Lvov, Mazur was the one closest to him. The way that the mathematicians worked in Lvov has become famous. They spent many hours thinking about mathematical problems in the Scottish Café. These sessions are described in [1]:-

You can see a picture of the Scottish Café at THIS LINK.Usually they began arriving between5and7pm - always occupying the same tables - and for the next several hours they worked with total concentration, covering the marble table tops with mathematical farmulas. But saying that they worked with total concentration is not completely accurate, as there was no meeting without jokes, heated discourse, shouting and drinking.

Ulam, in [2], describes the particular way that the collaboration between Mazur and Banach in the Scottish Café worked:-

It was in the Scottish Café that the famousWe discussed problems proposed right there, often with no solution evident even after several hours of thinking. The next day Banach was likely to appear with several small sheets of paper containing outlines of proofs he had completed. If they were not polished or even not quite correct, Mazur would frequently put them into a more satisfactory form.

*Scottish Book*consisting of open questions posed by the mathematicians working there came into being. Mazur contributed 24 problems to the book with himself as the sole author, and a further 19 problems jointly contributed with others such as Banach.

It was not only Banach with who Mazur collaborated but others including Ulam. In [2] Ulam describes their collaborations on function spaces:-

During the 1930s Mazur was an active member of the Polish Communist Party. This would stand him in good stead when the Communists came to power after the war. Mazur's habilitation thesis was submitted in 1936. After the award of his habilitation, Mazur taught at Lvov until 1946.We found a solution to a problem involving infinite dimensional vector spaces. The theorem we proved - that a transformation preserving distances is linear - is now part of the standard treatment of the geometry of function spaces. We wrote a paper which was published in Comptes Rendus ... It was Mazur ... who introduced me to certain large phases of mathematical thinking and approaches. From him I learned much about the attitudes and psychology of research.

In [2] Ulam recalls a conversation he had with Mazur in the summer of 1939 about whether he thought there would be a war:-

Although it is not known for certain how thePeople in general expected another crisis in the style of Munich and were not prepared for the coming World War. Mazur said "The possibility of a World War is real. What are we going to do with the Scottish Book and our joint unpublished papers? You are going to the United States, and most surely you are going to be safe. In case the city is bombed, I will pack the manuscripts and the Book in a chest and bury it". We even decided on an exact place - next to the goal post at the soccer pitch on the city outskirts. I do not know if that's what happened.

*Scottish Book*survived the war, we do know that it was brought to Wrocław by Banach's wife and it ended up in the possession of his son.

There is a charming story about one of the most famous of the problems in the *Scottish Book* which was posed by Mazur. This was problem number 153, which Mazur inserted into the *Book* on 6 November 1936. The problem asked (although not in these words) about the existence of Schauder bases in separable Banach spaces. As with many of the problems in the *Scottish Book* the proposer would offer a prize for their solution. Prizes offered included wine, spirits, or a meal in Cambridge but Mazur offered a live goose as the prize for this particular problem. Per Enflo showed in 1972 that the problem had a negative solution and, while in Warsaw lecturing on his solution, Mazur presented him with his prize, the live goose!

From 1948 Mazur worked at the University of Warsaw. Then, given his earlier involvement with the Polish Communist Party, Mazur became a high official in the science establishment.

Mazur made important contributions to geometrical methods in linear and non-linear functional analysis and to the study of Banach algebras. The mean ergodic theorem in Banach spaces was announced by Mazur in 1932 but a proof does not appear in print until 1938 when Yosida and by Kakutani published the result.

He was also interested in summability theory, infinite games and recursive functions. Ulam recounts in [2] how Mazur gave the first examples of infinite games in the Scottish Café in Lvov.

The author of [4] points out how many of Mazur's original contributions are not explicitly identified as such but appear in print only as remarks in Banach's *Théorie des operations lineaires*. For example, the weak-basis theorem, due to Mazur, is given by Banach in his book but no proof appears.

In 1978 Mazur was honoured by receiving honorary life membership in the Polish Mathematical Society. The article [6] presents the depth of Mazur's functional-analytic contribution which led to this honour. In 1980 the University of Warsaw awarded Mazur an honorary doctorate. The award was made in recognition that he was a leading Polish mathematician and a cofounder with Banach of the Polish School of Functional Analysis. In the article [3] Mazur's reply at the awarding ceremony is given. In his speach Mazur concentrated on his activities at the University of Lvov during World War II.

**Article by:** *J J O'Connor* and *E F Robertson*

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