After completing his secondary education, he befriended Hugo Steinhaus, with whom he established the Polish Mathematical Society in 1919 and later published the scientific journal Studia Mathematica.

Stefan Banach was born on 30 March 1892 at St. Lazarus General Hospital in Kraków, then part of the Austro-Hungarian Empire, into a Góral Roman Catholic family,[4] and was subsequently baptised by his father.

Military regulations did not permit soldiers of Stefan Greczek's rank to marry; he was a private and as the mother was too poor to support the child, the couple decided that he should be reared by family and friends.

[9] Stefan spent the first few years of his life with his grandmother, but when she was taken ill, Greczek arranged for his son to be raised by Franciszka Płowa and her niece Maria Puchalska in Kraków.

As Banach had to earn money to support his studies it was not until 1914 that he finally, at age 22, passed his high school graduation exams.

He attended some lectures at the Jagiellonian University at that time, including those of the famous Polish mathematicians Stanisław Zaremba and Kazimierz Żorawski, but little is known of that period of his life.

[17] In 1916, in Kraków's Planty park, Banach encountered Professor Hugo Steinhaus, one of the renowned mathematicians of the time.

The doctoral thesis, accepted by King John II Casimir University of Lwów in 1920[21] and published in 1922,[22] included the basic ideas of functional analysis, which was soon to become an entirely new branch of mathematics.

Banach, from 1939 a corresponding member of the Academy of Sciences of Ukraine, and on good terms with Soviet mathematicians,[5] had to promise to learn Ukrainian to be allowed to keep his chair and continue his academic activities.

[26] After the German takeover of Lwów in 1941 during Operation Barbarossa, all universities were closed and Banach, along with many colleagues and his son, was employed as a lice feeder at Professor Rudolf Weigl's Typhus Research Institute.

Employment in Weigl's Institute provided many unemployed university professors and their associates protection from random arrest and deportation to Nazi concentration camps.

After the Soviet Red Army recaptured Lviv 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 deporting Poles from annexed formerly Polish eastern territories, Banach began preparing to leave the city and settle in Kraków, Poland, where he had been promised a chair at the Jagiellonian University.

[28] The theory of what came to be known as Banach spaces had antecedents in the work of the Hungarian mathematician Frigyes Riesz (published in 1916) and contemporaneous contributions from Hans Hahn and Norbert Wiener.

[30] Stefan Banach is the patron of a number of schools and streets including in Warsaw, Lviv, Świdnica, Toruń and Jarosław.

[32] In 2016, a commemorative bench featuring Banach and Otto Nikodym was unveiled in Kraków's Planty Park on the 100th anniversary of the conversation the two mathematicians held when they first met Hugo Steinhaus, which proved instrumental in the development of his scientific career.

[33] In 2021, one of the episodes of Polish documentary TV series Geniusze i marzyciele (Geniuses and Dreamers) aired on TVP1 and TVP Dokument channels was devoted to Stefan Banach.