Lwów School of Mathematics in the context of "Stefan Banach"

Lviv (/ləˈviːv/ lə-VEEV or /ləˈviːf/ lə-VEEF; Ukrainian: Львів [ˈlʲviu̯] ; Polish: Lwów [ˈlvuf] ; see below for other names) is the largest city in western Ukraine, as well as the fifth-largest city in Ukraine, officially with a population of 723,403 (2025 estimate). It serves as the administrative centre of Lviv Oblast and Lviv Raion, and is one of the main cultural centres of Ukraine. Lviv also hosts the administration of Lviv urban hromada. It was named after Leo I of Galicia, the eldest son of Daniel, King of Ruthenia.

Lviv (then Lwów) emerged as the centre of the historical regions of Red Ruthenia and Galicia in the 14th century, superseding Halych, Chełm, Belz, and Przemyśl. It was the capital of the Kingdom of Galicia–Volhynia from 1272 to 1340, when it went to King Casimir III the Great of Poland in a war of succession. In 1356, Casimir the Great granted it town rights. From 1434, it was the regional capital of the Ruthenian Voivodeship in the Kingdom of Poland. In 1772, after the First Partition of Poland, the city became the capital of the Habsburg semi-autonomous Polish-dominated Kingdom of Galicia and Lodomeria. From 1918, between the wars, the city was the centre of the Lwów Voivodeship in the Second Polish Republic. There it flourished in culture, industry and academia such as the Lwów School of Mathematics, the Lwów Historical School (Polish: lwowska szkoła historyczna) and the Lwów School of Economics. After the German-Soviet invasion of Poland in 1939, the massacre of Lwów professors took place, and Lwów was eventually annexed by the Soviet Union.

Lviv (/ləˈviːv/ lə-VEEV or /ləˈviːf/ lə-VEEF; Ukrainian: Львів [ˈlʲviu̯] ; see below for other names; Polish: Lwów Polish pronunciation: [lvuf] ) is the largest city in western Ukraine, as well as the fifth-largest city in Ukraine, officially with a population of 723,403 (2025 estimate). It serves as the administrative centre of Lviv Oblast and Lviv Raion, and is one of the main cultural centres of Ukraine. Lviv also hosts the administration of Lviv urban hromada. It was named after Leo I of Galicia, the eldest son of Daniel, King of Ruthenia.

Lviv (then Lwów) emerged as the centre of the historical regions of Red Ruthenia and Galicia in the 14th century, superseding Halych, Chełm, Belz, and Przemyśl. It was the capital of the Kingdom of Galicia–Volhynia from 1272 to 1340, when it went to King Casimir III the Great of Poland in a war of succession. In 1356, Casimir the Great granted it town rights. From 1434, it was the regional capital of the Ruthenian Voivodeship in the Kingdom of Poland. In 1772, after the First Partition of Poland, the city became the capital of the Habsburg semi-autonomous Polish-dominated Kingdom of Galicia and Lodomeria. From 1918, between the wars, the city was the centre of the Lwów Voivodeship in the Second Polish Republic. There it flourished in culture, industry and academia such as the Lwów School of Mathematics, the Lwów Historical School (Polish: lwowska szkoła historyczna) and the Lwów School of Economics. After the German-Soviet invasion of Poland in 1939, the massacre of Lwów professors took place, and Lwów was eventually annexed by the Soviet Union.

Hugo Dyonizy Steinhaus (English: /ˈhjuːɡoʊ ˈstaɪnˌhaʊs/ HYOO-goh STYNE-howss, Polish: [ˈxuɡɔ ˈʂtajnxaws]; 14 January 1887 – 25 February 1972) was a Polish mathematician and educator. Steinhaus obtained his PhD under David Hilbert at Göttingen University in 1911 and later became a professor at the Jan Kazimierz University in Lwów (now Lviv, Ukraine), where he helped establish what later became known as the Lwów School of Mathematics. He is credited with "discovering" mathematician Stefan Banach, with whom he gave a notable contribution to functional analysis through the Banach–Steinhaus theorem. After World War II Steinhaus played an important part in the establishment of the mathematics department at Wrocław University and in the revival of Polish mathematics from the destruction of the war.

Author of around 170 scientific articles and books, Steinhaus has left his legacy and contribution in many branches of mathematics, such as functional analysis, geometry, mathematical logic, and trigonometry. Notably he is regarded as one of the early founders of game theory and probability theory, which led to later development of more comprehensive approaches by other scholars.

The Scottish Book (Polish: Księga Szkocka) was a thick notebook used by mathematicians of the Lwów School of Mathematics in Poland for jotting down problems meant to be solved. The notebook was named after the "Scottish Café" where it was kept.

Originally, the mathematicians who gathered at the cafe would write down the problems and equations directly on the cafe's marble table tops, but these would be erased at the end of each day, and so the record of the preceding discussions would be lost. The idea for the book was most likely originally suggested by Stefan Banach's wife, Łucja Banach. Stefan or Łucja Banach purchased a large notebook and left it with the proprietor of the cafe.

The Scottish Café (Polish: Kawiarnia Szkocka) was a café in Lwów, Poland (now Lviv, Ukraine) where, in the 1930s and 1940s, mathematicians from the Lwów School of Mathematics collaboratively discussed research problems, particularly in functional analysis and topology.

Stanisław Ulam recounts that the tables of the café had marble tops, so they could write in pencil, directly on the table, during their discussions. To keep the results from being lost, and after becoming annoyed with their writing directly on the table tops, Stefan Banach's wife provided the mathematicians with a large notebook, which was used for writing the problems and answers and eventually became known as the Scottish Book. The book—a collection of solved, unsolved, and even probably unsolvable problems—could be borrowed by any of the guests of the café. Solving any of the problems was rewarded with prizes, with the most difficult and challenging problems having expensive prizes (during the Great Depression and on the eve of World War II), such as a bottle of fine brandy.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, inner product, norm, or topology) and the linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous or unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

The usage of the word functional as a noun goes back to the calculus of variations, implying a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. However, the general concept of a functional had previously been introduced in 1887 by the Italian mathematician and physicist Vito Volterra. The theory of nonlinear functionals was continued by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis further developed by Riesz and the group of Polish mathematicians around Stefan Banach.