Editing Banach–Alaoglu theorem (section)

{{Short description|Theorem in functional analysis}}
In [[functional analysis]] and related branches of [[mathematics]], the '''Banach–Alaoglu theorem''' (also known as '''Alaoglu's theorem''') states that the [[Closed set|closed]] [[Ball (mathematics)|unit ball]] of the [[dual space]] of a [[normed vector space]] is [[Compact space|compact]] in the [[weak* topology]].<ref>{{harvnb|Rudin|1991}}, Theorem 3.15.</ref> 
A common proof identifies the unit ball with the weak-* topology as a closed subset of a [[Cartesian product|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.