Stefan Banach - Contributions

Contributions

Banach's dissertation, completed in 1920 and published in 1922, formally axiomatized the concept of a complete normed vector space and laid the foundations for the area of functional analysis. In this work Banach called such spaces "class E-spaces", but in his 1932 book, Théorie des opérations linéaires, he changed terminology and referred to them as "spaces of type B", which most likely contributed to the subsequent eponymous naming of these spaces after him. 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. For a brief period in fact, complete normed linear spaces where referred to as "Banach-Wiener" spaces in mathematical literature, based on terminology introduced by Wiener himself. However, because Wiener's work on the topic was limited, the established name became just Banach spaces.

Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students (for example in the Banach–Schauder theorem) and other mathematicians (in particular Bouwer and Poincaré and Birkhoff). The theorem did not require linearity of the space, and applied to any Cauchy space (complete metric space).

The Hahn–Banach theorem, is one of fundamental theorems of functional analysis.

• Banach–Tarski paradox
• Banach–Steinhaus theorem
• Banach–Alaoglu theorem
• Banach–Stone theorem