Editing Banach–Alaoglu theorem (section)

==History==

According to Lawrence Narici and Edward Beckenstein, the Alaoglu theorem is a “very important result—maybe {{em|the}} most important fact about the [[weak-* topology]]—[that] echos throughout functional analysis.”{{sfn|Narici|Beckenstein|2011|pp=235-240}} 
In 1912, Helly proved that the unit ball of the continuous dual space of <math>C([a, b])</math> is countably weak-* compact.{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
In 1932, [[Stefan Banach]] proved that the closed unit ball in the continuous dual space of any [[Separable metric space|separable]] [[normed space]] is sequentially weak-* compact (Banach only considered [[sequential compactness]]).{{sfn|Narici|Beckenstein|2011|pp=225-273}} 
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.{{sfn|Narici|Beckenstein|2011|pp=235-240}} 

The '''Bourbaki–Alaoglu theorem''' is a generalization<ref>{{harvnb|Köthe|1983}}, Theorem (4) in §20.9.</ref><ref>{{harvnb|Meise|Vogt|1997}}, Theorem 23.5.</ref> of the original theorem by [[Nicolas Bourbaki|Bourbaki]] to [[dual topology|dual topologies]] on [[locally convex space]]s. 
This theorem is also called the '''Banach–Alaoglu theorem''' or the '''weak-* compactness theorem''' and it is commonly called simply the '''Alaoglu theorem'''.{{sfn|Narici|Beckenstein|2011|pp=235-240}}