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.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.
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 :-
Usually they began arriving between 5 and 7 pm - 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 , describes the particular way that the collaboration between Mazur and Banach in the Scottish Café worked:-
We 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.It was in the Scottish Café that the famous 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  Ulam describes their collaborations on function spaces:-
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.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.
In  Ulam recalls a conversation he had with Mazur in the summer of 1939 about whether he thought there would be a war:-
People 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.Although it is not known for certain how the 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  how Mazur gave the first examples of infinite games in the Scottish Café in Lvov.
The author of  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  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  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