banach

Banach is a surname of Polish origin most associated with Stefan Banach (1892–1945), a leading mathematician and one of the founders of functional analysis. A central member of the Lwów School of Mathematics, Banach contributed to the development of modern analysis while working at the University of Lwów and with colleagues such as Hugo Steinhaus and Stanisław Ulam. His work helped formalize the theory of infinite-dimensional spaces and operators.

Many fundamental objects and results in functional analysis bear his name. A Banach space is a complete normed vector space, the natural setting for analysis in infinite dimensions. The Banach fixed-point theorem provides conditions under which a contraction on a complete metric space has a unique fixed point, with numerous applications. Banach also helped develop the theory of Lp spaces and Banach algebras in functional analysis. The Banach–Steinhaus theorem (Uniform Boundedness Principle) and the Banach–Mazur distance are named after him and his collaborators. He coauthored work with Steinhaus that contributed to the Banach–Steinhaus theorem, and his collaboration with the Lwów School helped established many foundational results.

In set-theoretic geometry, Banach is also associated with the Banach–Tarski paradox, proven with Alfred Tarski, which states that a solid ball in three-dimensional space can be partitioned and reassembled into two identical copies of the original ball, using the axiom of choice. Banach’s work and the Lwów School had a lasting influence on analysis and functional analysis worldwide.