Stefan Banach

Stefan Banach (1892–1945) was a Polish mathematician widely regarded as one of the most significant figures in 20th-century mathematics. He helped create modern functional analysis and was a founding member of the Lwów School of Mathematics. His landmark 1932 book, *Théorie des opérations linéaires*, was the first comprehensive work on functional analysis.

Banach was born on 30 March 1892 in a Kraków hospital, then part of the Austro-Hungarian Empire, to a Góral Roman Catholic family. His parents were Stefan Greczek, a soldier stationed in Kraków, and Katarzyna Banach, a domestic helper from the Podhale region. Because military rules prevented Greczek from marrying, Banach took his mother’s surname and his father’s first name. His mother could not support him, so he was raised by his grandmother until she fell ill, then by Franciszka Płowa and her niece Maria Puchalska in Kraków. Banach considered Franciszka his foster mother and Maria his sister. A French intellectual named Juliusz Mien, a friend of the family, tutored Banach in French and likely encouraged his early interest in math.

At age 10, Banach entered Kraków’s IV Gymnasium, a humanities-focused school. He and his friend Witold Wiłkosz, also a future mathematician, spent breaks and after-school time solving math problems. Banach credited his math teacher Dr. Kamil Kraft with sparking his passion for the subject, though he sometimes got low grades—he failed Greek his first semester—and later criticized the school’s math instruction. After earning his matura in 1910 at 18, he moved to Lwów to study engineering at the Lwów Polytechnic, believing there was nothing new left to discover in math. He also attended Jagiellonian University part-time. To fund his studies, he worked various jobs, and he didn’t pass his final high school exams until 1914, at age 22.

When World War I began, Banach was exempted from military service because he was left-handed and had poor vision. As the Russian Army advanced on Lwów, he returned to Kraków, where he tutored, worked in a bookstore, and supervised a road-building crew. He attended some lectures at Jagiellonian University by mathematicians Stanisław Zaremba and Kazimierz Żorawski, but little else is known from this period.

In 1916, Banach met Professor Hugo Steinhaus in Kraków’s Planty park. Steinhaus overheard Banach mention the Lebesgue integral—still a novel concept—and approached him, also meeting Otto Nikodym. Impressed by the self-taught Banach, Steinhaus asked him to solve some tough problems; Banach did so within a week. They soon published their first joint paper, *On the Mean Convergence of Fourier Series*. Along with Nikodym and other Kraków mathematicians, they founded a society that officially became the Polish Mathematical Society on 2 April 1919. Steinhaus also introduced Banach to his future wife, Łucja Braus.

Banach earned an assistantship at the Lwów Polytechnic in 1920, became a professor in 1922, and joined the Polish Academy of Learning in 1924. He co-founded the Lwów School of Mathematics, a group of leading Polish mathematicians active between the world wars. Many mathematical concepts bear his name: Banach spaces, Banach algebras, Banach measures, the Banach–Tarski paradox, the Hahn–Banach theorem, the Banach–Steinhaus theorem, the Banach–Mazur game, the Banach–Alaoglu theorem, the Banach-Saks property, and the Banach fixed-point theorem.