ProofWiki:Mathematicians/Stefan Banach

Polish mathematician who founded the modern field of functional analysis.

Most famous for his collaborative paper with Alfred Tarski in 1924, in which the Banach-Tarski Paradox was raised. This demonstrated that a contra-intuitive truth could be deduced from the ZFC axioms of set theory, specifically, by assuming the truth of the Axiom of Choice. Impassioned controversy rages to this day.

Nationality

Polish

History

• Born: 30 March 1892, Ostrowsko, near Kraków, Austria-Hungary (now Poland)
• Died: 31 Aug 1945, Lvov, (now Ukraine)

Theorems and Definitions

• Amenable Banach Algebra
• Banach-Alaoglu Theorem (with Leonidas Alaoglu)
• Banach Algebra
• Banach Bundle
• Banach Fixed-Point Theorem
• Banach Function Algebra
• Banach Integral
• Banach Limit
• Banach Manifold
• Banach's Matchbox Problem
• Banach-Mazur Game (with Stanisław Mazur)
• Banach-Mazur Theorem (with Stanisław Mazur)
• Banach Measure
• Banach-Schauder Theorem (with Juliusz Paweł Schauder)
• Banach Space
• Banach-Steinhaus Theorem (with Hugo Steinhaus)
• Banach-Stone Theorem (with Marshall Harvey Stone)
• Banach-Tarski Paradox (with Alfred Tarski)
• Closed Range Theorem (proved by Banach in 1932: Théorie des opérations linéaires)
• Hahn-Banach Theorem (with Hans Hahn)

Books and Papers

• 1922: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales (Fund. Math. Vol. 3: 133 – 181) (in which Banach Fixed-Point Theorem is presented)
• 1924: Sur la décomposition des ensembles de points (with Alfred Tarski)
• 1929: Rachunek różniczkowy i całkowy, tom I (Differential and Integral Calculus, vol. 1)
• 1930: Rachunek różniczkowy i całkowy, tom II (Differential and Integral Calculus, vol. 2)
• 1931: Über die Baire'sche Kategorie gewisser Funktionenmengen (Studia Math. Vol. 3: 174 – 179)
• 1931: Teoria operacji. Tom I. Operacje liniowe (Theory of operations, vol. 1: Linear operations)
• 1932: Théorie des opérations linéaires (Theory of Linear Operations) (in which the concept now known as the Banach space was formulated)
• 1938: Mechanika w zakresie szkół akademickich, Monografie Matematyczne 8 (Mechanics for Academic Schools, Mathematical Monographs 8)

Also see

• John J. O'Connor and Edmund F. Robertson: "Stefan Banach": MacTutor History of Mathematics archive