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Home > Banach-Alaoglu theorem

The Banach-Alaoglu theorem (also known as Alaoglu's Theorem) states that the closed unit ball of the dual space of a normed vector space is compact in the weak* topology. A common proof identifies the unit ball with the weak* topology as a closed subset of a product of compact sets with the product topology. As a consequence of Tychonoff's theorem, this product, and hence the unit ball within, is compact.

A proof of this theorem for separable normed vector spaces was published in 1932 by Stefan Banach, and the first proof for the general case was published in 1940 by the Turkish mathematician Leonidas Alaoglu .

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Banach spaceFunctional analysis In mathematics, Banach spaces named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. Banach spaces are typically infinite-dimensional spaces containing functions. Definition Banach s Banach algebraIn functional analysis, a Banach algebra named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. The algebra multiplication and the Banach space norm are required to be related Banach fixed point theoremTopology Mathematical analysis Theorems The Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive metho Banach-Tarski paradoxFirst stated by Stefan Banach and Alfred Tarski in 1924, the Banach-Tarski paradox is the famous "doubling the ball" paradox, which states that by using the axiom of choice it is possible to take a solid ball in 3-dimensional space, cut it up into finitel Banach-Mazur gameIn mathematics, in particular in general topology and set theory, a Banach-Mazur game is a game played between two players, trying to pin down elements in a set (space). The concept of a Banach-Mazur game is closely related to the technical concept of Bai Banach-Alaoglu theoremFunctional analysis The Banach-Alaoglu theorem (also known as Alaoglu's Theorem states that the closed unit ball of the dual space of a normed vector space is compact in the weak topology. A common proof identifies the unit ball with the weak topology as Banachiewicz (crater)