As Stan Wagon points out at the end of his monograph, the Banach–Tarski paradox has been more significant for its role in pure mathematics than for foundational questions: it motivated a fruitful new direction for research, the amenability of groups, which has nothing to do with the foundational questions.
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Banach and Tarski explicitly acknowledge Giuseppe Vitali's 1905 construction of the set bearing his name, Hausdorff's paradox (1914), and an earlier (1923) paper of Banach as the precursors to their work.
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In mathematics, Banach measure in measure theory may mean a real-valued function on an algebra of all subsets of a set (for example, all subsets of the plane), by means of which a rigid, finitely additive area can be defined for every set, even when a set does not have a true geometric area.
It follows from the Hahn–Banach separation theorem that the weak topology is Hausdorff, and that a norm-closed convex subset of a Banach space is also weakly closed.
In functional analysis, the Dunford–Pettis property, named after Nelson Dunford and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous.
Compact sets in Banach spaces may also carry natural measures: the Hilbert cube, for instance, carries the product Lebesgue measure.
Kōmura's theorem, result on the differentiability of absolutely continuous Banach space-valued functions
In one case, David Banach of New Jersey was charged under federal Patriot Act anti-terrorism laws, after he allegedly shone a laser pointer at aircraft.
He completed his PhD, entitled "La propriété du point fixe dans les espaces de Banach et les espaces Metriques", at the Pierre-and-Marie-Curie University in May 1987 under the supervision of Gilles Godefroy.
In 1989, A. K. Dewdney published a letter from his friend Arlo Lipof in the Computer Recreations column of the Scientific American where he describes an underground operation "in a South American country" of doubling gold balls using the Banach–Tarski paradox.
Norman Leto (Łukasz Banach, born in 1980, Bochnia) An artist, self-educated in the field of video, film and new media.
Ed and Lou Banach, 1984 Summer Olympics wrestling gold medalists lived in Port Jervis and graduated from Port Jervis Senior High School.
# Banach's extension theorem which is used to prove one of the most fundamental results in functional analysis, the Hahn–Banach theorem