Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Banach–Alaoglu theorem
Summary
In functional analysis and related branches of mathematics, 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.
This theorem has applications in physics when one describes the set of states of an algebra of observables, namely that any state can be written as a convex linear combination of so-called pure states.
According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a “very important result—maybe most important fact about the weak-* topology—[that] echos throughout functional analysis.”
In 1912, Helly proved that the unit ball of the continuous dual space of is countably weak-* compact.
In 1932, Stefan Banach proved that the closed unit ball in the continuous dual space of any separable normed space is sequentially weak-* compact (Banach only considered sequential compactness).
The proof for the general case was published in 1940 by the mathematician Leonidas Alaoglu.
According to Pietsch [2007], there are at least twelve mathematicians who can lay claim to this theorem or an important predecessor to it.
The Bourbaki–Alaoglu theorem is a generalization of the original theorem by Bourbaki to dual topologies on locally convex spaces.
This theorem is also called the Banach–Alaoglu theorem or the weak-* compactness theorem and it is commonly called simply the Alaoglu theorem.
Topological vector space#Dual spaceDual system and Polar set
If is a vector space over the field then will denote the algebraic dual space of and these two spaces are henceforth associated with the bilinear defined by
where the triple forms a dual system called the .
If is a topological vector space (TVS) then its continuous dual space will be denoted by where always holds.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.