A self-taught mathematics prodigy, Banach was the founder of modern functional analysis and a founder of the Lwów School of Mathematics. Among his most prominent achievements was the 1932 book, Théorie des opérations linéaires (Theory of Linear Operations), the first monograph on the general theory of linear-metric space.
Unusually, Stefan's surname was that of his mother instead of his father, though he received his father's given name. Since Banach's father was a private and was prevented by military regulations from marrying, and the mother was too poor to support the child, the couple decided that he be reared by a friend of Banach's father, the owner of a Kraków laundry.
Banach's mother left him after baptizing him when he was four days old. Her name on the birth certificate is Katarzyna Banach. Later in life Banach would ask his father to tell him his mother's actual identity but would only be told that he had taken an oath of secrecy about it. Stefan Greczek would go on to marry twice and have a son by his first wife and four children by the second. He would pay for his son's education and be the only relative whom Banach would ever know personally.
Already as a student at Kraków's Gymnasium no. IV, Banach became known as a prodigy. In 1906, aged 14, he was studying higher mathematics, and two years later he had started in on several languages, western and eastern. After obtaining his matura at age 18 in 1910, Banach went with his friend Witold Wiłkosz to Lviv, then the capital of Galicia, intending to enroll in engineering at the Lwów Polytechnic. However, as Banach had to earn money to support his studies, it was not until 1914 that he finally, at age 22, passed his half-diploma exams.
When World War I broke out, Banach was excused from military service due to his left-handedness and poor vision. When the Russian Army opened its offensive toward Lviv, Banach left for Kraków, to spend the rest of the war there and in other Galician towns. He made his living tutoring at local gymnasiums and working in a bookshop. He may have attended lectures at the Jagiellonian University, but little is known of that period in his life.
In 1916, in Cracow's Planty gardens, Banach encountered Professor Hugo Steinhaus, one of the most renowned mathematicians of the age. Steinhaus became fascinated with the self-taught young mathematician. The encounter resulted in a long-lasting collaboration and friendship. It was also through Steinhaus that Banach met his future wife, Łucja Braus.
Steinhaus introduced Banach to academic circles and substantially accelerated his career. After Poland regained independence, in 1920 Banach was given an assistantship at Kraków's Jagiellonian University. Steinhaus' backing also allowed him to receive a doctorate without actually graduating from a university. The doctoral thesis, accepted by Lwów University and published in 1922, included the basic ideas of functional analysis, which was soon to become an entirely new branch of mathematics. The thesis was widely discussed in academic circles and allowed him in 1922 to become a professor at the Lwów Polytechnic. Initially an assistant to Professor Antoni Łomnicki, in 1927 Banach received his own chair. In 1924 he was also accepted as a member of the Polish Academy of Learning. At the same time, from 1922, Banach also headed the second Chair of Mathematics at Lwów University.
Young and talented, Banach gathered around him a large group of mathematicians. The group, meeting in the Scottish Café, soon gave birth to the "Lwów School of Mathematics." In 1929 the group began publishing its own journal, Studia Mathematica, devoted primarily to Banach's field of study — functional analysis. Around that time, Banach also began working on his best-known work, the first monograph on the general theory of linear-metric space. First published in Polish in 1931, the following year it was also translated into French and gained wider recognition in European academic circles. The book was also the first in a long series of mathematics monographs edited by Banach and his circle.
Following the invasion of Poland by Nazi Germany and the Soviet Union, Lwów came under the control of the Soviet Union. Banach, from 1939 a corresponding member of the Academy of Sciences of Ukraine, and on good terms with Soviet mathematicians, had to promise to learn Ukrainian to be allowed to keep his chair and continue his academic activities. Following the German takeover of Lviv in 1941, all universities were closed and Banach, along with many colleagues and his son, was forced to eke out a living feeding lice with his blood at Professor Rudolf Weigl's Typhus Research Institute.
After the Red Army captured Lvov in the Lvov–Sandomierz Offensive of 1944, Banach returned to the University and helped re-establish it after the war years. However, because the Soviets were removing Poles from annexed formerly-Polish territories, Banach began preparing to leave the city and settle in Kraków, Poland, where he had been promised a chair at the Jagiellonian University. He was also considered a candidate for Minister of Education of Poland. In January 1945, however, he was diagnosed with lung cancer and was allowed to stay in Lviv. He died on August 31, 1945, aged 53. His funeral at the Lychakiv Cemetery turned into a patriotic demonstration by the Poles who still remained in the city.
Banach's most influential work was Théorie des opérations linéaires (Theory of Linear Operations, 1932). In it he formulated the concept now known as "Banach space," and proved many fundamental theorems of functional analysis.
He was also one of the founders and editors of the journal, Studia Mathematica.
Hugo Steinhaus said of Banach: