Steinhaus studied for one year in Lvov, spent one term in Munich but then spent five years studying mathematics at the University of Göttingen. There he was influenced by an amazingly strong group of mathematicians including Felix Bernstein, Carathéodory, Courant, Herglotz, Hilbert, Klein, Koebe, Edmund Landau Landau (although he only arrived in Göttingen after Steinhaus had been there three years), Runge, Toeplitz, and Zermelo. For his doctorate Steinhaus studied under Hilbert's supervision. He was awarded his doctorate, with distinction, for a dissertation Neue Anwendungen des Dirichlet'schen Prinzips Ⓣ in 1911.
The main influence on the direction that Steinhaus's research would take was none of the major mathematical figures at Göttingen but rather the influence came from Lebesgue. Steinhaus studied Lebesgue's two major books Leçons sur l'intégration et la recherche des fonctions primitives Ⓣ (1904) and Leçons sur les séries trigonmétriques Ⓣ (1906) around 1912 after completing his doctorate.
After military service in the Polish Legion at the beginning of World War I, Steinhaus lived in Kraków. He relates in  how, despite the war in 1916, it was safe to walk in Kraków:-
During one such walk I overheard the words "Lebesgue measure". I approached the park bench and introduced myself to the two young apprentices of mathematics. They told me they had another companion by the name of Witold Wilkosz, whom they extravagantly praised. The youngsters were Stefan Banach and Otto Nikodym. From then on we would meet on a regular basis, and ... we decided to establish a mathematical society.The mathematical society which Steinhaus proposed was started as the Mathematical Society of Kraków and, shortly after the war ended, it became the Polish Mathematical Society. Steinhaus described the beginnings of the new mathematical society in  in a passage which tells us quite a lot about his life in Kraków at the time:-
As initiator of the idea, I made my room available for meetings and, as the first step in preparations, nailed an oilcloth blackboard to the wall. When the French manager of the boarding house saw what I had done, she was terrified - what was the proprietor going to say? I calmed her down reminding her that the owner of the building was my uncle's brother-in-law, and she forgave my transgression. However, I had made a mistake. Mr L took the position of a traditional, hard-nosed landlord and was unmoved by the lofty goal the blackboard was supposed to serve. The society expanded - it was the first ray of light of this kind in Poland.Also at this time Steinhaus started a collaboration with Banach and their first joint work was completed in 1916. Steinhaus took up an appointment as an assistant at the Jan Kazimierz University in Lvov and, around 1920, he was promoted to Extraordinary Professor. Banach was by this time on the staff at Lvov and the school rapidly grew in importance. Kac, who was a student of Steinhaus in Lvov during the 1930s, described the influence of Lebesgue's work on the Lvov school:-
The influence of Lebesgue on the Lvov school was very direct. The school, founded ... by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory. There is no doubt that none of these theories would have achieved today's level of prominence without an essential understanding of the Lebesgue measure and integral. On the other hand, the ideas of Lebesgue measure and integral found their most striking and fruitful applications there in Lvov.Steinhaus was the main figure in the Lvov School up till 1941. In 1923 he published in Fundamenta Mathematicae the first rigorous account of the theory of tossing coins based on measure theory. In 1925 he was the first to define and discuss the concept of strategy in game theory. Steinhaus published his second joint paper with Banach in 1927 Sur le principe de la condensation des singularités Ⓣ. In 1929, together with Banach, he started a new journal Studia Mathematica and Steinhaus and Banach became the first editors. The editorial policy was:-
... to focus on research in functional analysis and related topics.Another important publishing venture in which Steinhaus was involved, begun in 1931, was a new series of Mathematical Monographs. The series was set up under the editorship of Steinhaus and Banach from Lvov and Knaster, Kuratowski, Mazurkiewicz, and Sierpiński from Warsaw. An important contribution to the series was a volume written by Steinhaus jointly with Kaczmarz in 1937, The theory of orthogonal series.
Steinhaus is best known for his book Mathematical Snapshots written in 1937. Kac, writing in  says:-
... to understand and appreciate Steinhaus's mathematical style, one must read (or rather look at) snapshots. ... designed to appeal to "the scientist in the child and the child in the scientist" ... it expresses, not always explicitly and at times even unconsciously, what Steinhaus thought mathematics is and should be. To Steinhaus mathematics was a mirror of reality and life much in the same way as poetry is a mirror, and he liked to "play" with numbers, sets, and curves, the way a poet plays with words, phrases, and sounds.Stark  describes Steinhaus lectures in Lvov:-
My class was guided by Professor Steinhaus. It was a very big class, and the analysis lecture was attended by over 220 students squeezed into a smallish and poorly ventilated lecture room, standing in the aisles, and sitting on the window sills. ... His figure, perched high on the podium by a small five by five foot blackboard dominated the crowded room. ... despite Steinhaus's attention to preparation, the lectures were too difficult for the average student.The mathematicians of the Lvov school did a great deal of mathematical research in the cafés of Lvov. The Scottish Café was the most popular with the mathematicians in general but not with Steinhaus who (according to Ulam):-
... usually frequented a more genteel tea shop that boasted the best pastry in Poland.
This was Ludwik Zalewski's Confectionery at 22 Akademicka Street. It was in the Scottish Café, however, that the famous Scottish Book consisting of open questions posed by the mathematicians working there came into being. Steinhaus, who sometimes joined his colleagues in the Scottish Café, contributed ten problems to the book, including the final one written on 31 May 1941 only days before the Nazi troops entered the town.
You can see a picture of the Scottish Café at THIS LINK.
When the prospect of war was looming in 1938, Steinhaus proposed Lebesgue for an honorary degree from Lvov. Steinhaus joked to Kac that :-
It will not be a bad record to leave behind, to have had Banach as the first and Lebesgue as the last doctoral candidate.The reception for Lebesgue, after the award of his degree, was held in the Scottish Café but only fifteen mathematicians attended, showing that the school of mathematics in Lvov had shrunk considerably due to the political situation. Steinhaus spent the war years from June 1941 hiding from the Nazis, suffering great hardships, going hungry most of the time but always thinking about mathematics :-
... even then his sharp restless mind was at work on a multitude of ideas and projects.In 1945 Steinhaus moved to the University of Wrocław but made many visits to universities in the United States including Notre Dame. Kac in  writes:-
... it was he who, perhaps more than any other individual, helped to raise Polish mathematics from the ashes to which it had been reduced by the Second World War to the position of new strength and respect which it now occupies.After the end of World War II the Scottish Book, which seems to have been preserved through the war by Steinhaus, was sent by him to Ulam in the United States. The book was translated into English by Ulam and published. Steinhaus, now in the University of Wrocław, decided that the tradition of the Scottish Book was too good to end. In 1946 he extended the tradition to Wrocław starting the New Scottish Book.
Let us finally examine some of Steinhaus's mathematical contributions which we have not mentioned above. In 1944 Steinhaus proposed the problem of dividing a cake into n pieces so that it is proportional (each person is satisfied with their share) and envy free (each person is satisfied nobody is receiving more than a fair share). For n = 2 the problem is trivial, one person cuts the cake, the other chooses their piece. Steinhaus found a proportional but not envy free solution for n = 3. An envy free solution to Steinhaus's problem for n = 3 was found in 1962 by John H Conway and, independently, by John Selfridge. For general n the problem was solved by Steven Brams and Alan Taylor in 1995.
Steinhaus's bibliography, see , contains 170 articles. He did important work on functional analysis, but he himself described his greatest discovery in this area as Stefan Banach. Some of Steinhaus's early work was on trigonometric series. He was the first to give some examples which would lead to marked progress in the subject. He gave an example of a trigonometric series which diverged at every point, yet its coefficients tended to zero. He also gave an example of a trigonometric series which converged in one interval but diverged in a second interval.
As we have noted above, other contributions by Steinhaus were on orthogonal series, probability theory, real functions and their applications. In particular he is associated with the theory of independent functions, arising from his work in probability theory, and he was the first to make precise the concepts of "independent" and "uniformly distributed". In addition to his famous book Mathematical Snapshots he also wrote the highly acclaimed One Hundred Problems ....
Article by: J J O'Connor and E F Robertson