A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories.

In his early years Banach was tutored by Juliusz Mien a French intellectual and friend of the Płowa family who had emigrated to Poland and supported himself with photography and translations of Polish literature into French.

Some of the notable mathematical concepts named after Banach include Banach spaces Banach algebras the Banach–Tarski paradox the Hahn–Banach theorem the Banach–Steinhaus theorem the Banach-Mazur game the Banach–Alaoglu theorem and the Banach fixed-point theorem. Banach was one of the founders of modern functional analysis and one of the original members of the Lwów School of Mathematics. He soon became a professor at Lwów Polytechnic and a member of the Polish Academy of Learning during this period.