Editing Banach–Alaoglu theorem (section)

==Relation to the axiom of choice and other statements==
{{See also|Krein–Milman theorem#Relation to other statements}}

The Banach–Alaoglu may be proven by using [[Tychonoff's theorem]], which under the [[Zermelo–Fraenkel set theory]] ('''ZF''') axiomatic framework is equivalent to the [[axiom of choice]]. 
Most mainstream functional analysis relies on '''ZF''' + the axiom of choice, which is often denoted by '''ZFC'''. 
However, the theorem does {{em|not}} rely upon the axiom of choice in the [[Separable space|separable]] case (see [[#Sequential Banach–Alaoglu theorem|above]]): in this case there actually exists a constructive proof. 
In the general case of an arbitrary normed space, the [[ultrafilter Lemma]], which is strictly weaker than the axiom of choice and equivalent to Tychonoff's theorem for compact {{em|Hausdorff}} spaces, suffices for the proof of the Banach–Alaoglu theorem, and is in fact equivalent to it.

{{anchor|Relation to the Hahn–Banach theorem}}The Banach–Alaoglu theorem is equivalent to the [[ultrafilter lemma]], which implies the [[Hahn–Banach theorem]] for [[real vector space]]s ('''HB''') but is not equivalent to it (said differently, Banach–Alaoglu is also strictly stronger than '''HB'''). 
However, the [[Hahn–Banach theorem]] is equivalent to the following weak version of the Banach–Alaoglu theorem for normed space<ref name=BellFremlin1972>{{cite journal|last1=Bell|first1=J.|last2=Fremlin|first2=David|title=A Geometric Form of the Axiom of Choice|journal=Fundamenta Mathematicae|date=1972|volume=77|issue=2|pages=167–170|doi=10.4064/fm-77-2-167-170|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm77/fm77116.pdf|access-date=26 Dec 2021}}</ref> in which the conclusion of compactness (in the [[weak-* topology]] of the closed unit ball of the dual space) is replaced with the conclusion of {{em|quasicompactness}} (also sometimes called {{em|convex compactness}}); 

{{Math theorem|name={{visible anchor|Weak version of Alaoglu theorem}}<ref name=BellFremlin1972 />|math_statement=
Let <math>X</math> be a normed space and let <math>B</math> denote the closed unit ball of its [[continuous dual space]] <math>X^{\prime}.</math> Then <math>B</math> has the following property, which is called ([[weak-* topology|weak-*]]) {{em|{{visible anchor|quasicompactness}}}} or {{em|{{visible anchor|convex compactness}}}}: whenever <math>\mathcal{C}</math> is a cover of <math>B</math> by {{em|convex}} [[weak-* topology|weak-* closed]] subsets of <math>X^{\prime}</math> such that <math>\{B \cap C : C \in \mathcal{C}\}</math> has the [[finite intersection property]], then <math>B \cap \bigcap_{C \in \mathcal{C}} C</math> is not empty.
}}

[[Compact space|Compactness]] implies [[#convex compactness|convex compactness]] because a topological space is compact if and only if every [[Family of sets|family]] of closed subsets having the [[finite intersection property]] (FIP) has non-empty intersection. 
The [[#convex compactness|definition of convex compactness]] is similar to this characterization of [[compact space]]s in terms of the FIP, except that it only involves those closed subsets that are also [[Convex set|convex]] (rather than all closed subsets).