Editing Hahn–Banach theorem (section)

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

{{anchor|Relation to axiom of choice}}The proof of the [[#Hahn–Banach dominated extension theorem|Hahn–Banach theorem for real vector spaces]] ('''HB''') commonly uses [[Zorn's lemma]], which in the axiomatic framework of [[Zermelo–Fraenkel set theory]] ('''ZF''') is equivalent to the [[axiom of choice]] ('''AC''').  It was discovered by [[Jerzy Łoś|Łoś]] and [[Czesław Ryll-Nardzewski|Ryll-Nardzewski]]{{sfn|Łoś|Ryll-Nardzewski|1951|pp=233–237}} and independently by [[Wilhelmus Luxemburg|Luxemburg]]{{sfn|Luxemburg|1962|p=}} that '''HB''' can be proved using the [[ultrafilter lemma]] ('''UL'''), which is equivalent (under '''ZF''') to the [[Boolean prime ideal theorem]] ('''BPI'''). '''BPI''' is strictly weaker than the axiom of choice and it was later shown that '''HB''' is strictly weaker than '''BPI'''.{{sfn|Pincus|1974|pp=203–205}}

The [[ultrafilter lemma]] is equivalent (under '''ZF''') to the [[Banach–Alaoglu theorem]],{{sfn|Schechter|1996|pp=766–767}} which is another foundational theorem in [[functional analysis]]. Although the Banach–Alaoglu theorem implies '''HB''',<ref name="Muger2020">{{cite book|last=Muger|first= Michael|title=Topology for the Working Mathematician|year=2020}}</ref> it is not equivalent to it (said differently, the Banach–Alaoglu theorem is strictly stronger than '''HB'''). 
However, '''HB''' is equivalent to [[Banach–Alaoglu theorem#Relation to the Hahn–Banach theorem|a certain weakened version of the Banach–Alaoglu theorem]] for normed spaces.<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> 
The Hahn–Banach theorem is also equivalent to the following statement:<ref>{{cite book|last=Schechter|first=Eric|title=Handbook of Analysis and its Foundations|page=620|author-link=Eric Schechter}}</ref>

:(∗): On every [[Boolean algebra (structure)|Boolean algebra]] {{mvar|B}} there exists a "probability charge", that is: a non-constant finitely additive map from <math>B</math> into <math>[0, 1].</math>

('''BPI''' is equivalent to the statement that there are always non-constant probability charges which take only the values 0 and 1.)

In '''ZF''', the Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set.<ref>{{cite journal|last1=Foreman|first1=M.|last2=Wehrung|first2=F.|year=1991|title=The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set|url=http://matwbn.icm.edu.pl/ksiazki/fm/fm138/fm13812.pdf|journal=Fundamenta Mathematicae|volume=138|pages=13–19|doi=10.4064/fm-138-1-13-19|doi-access=free}}</ref>  Moreover, the Hahn–Banach theorem implies the [[Banach–Tarski paradox]].<ref>{{cite journal|last=Pawlikowski|first=Janusz|year=1991|title=The Hahn–Banach theorem implies the Banach–Tarski paradox|journal=Fundamenta Mathematicae|volume=138|pages=21–22|doi=10.4064/fm-138-1-21-22|doi-access=free}}</ref>

For [[Separable space|separable]] [[Banach space]]s, D. K. Brown and S. G. Simpson proved that the Hahn–Banach theorem follows from WKL<sub>0</sub>, a weak subsystem of [[second-order arithmetic]] that takes a form of [[Kőnig's lemma]] restricted to binary trees as an axiom.  In fact, they prove that under a weak set of assumptions, the two are equivalent, an example of [[reverse mathematics]].<ref>{{cite journal|last1=Brown|first1=D. K.|last2=Simpson|first2=S. G.|year=1986|title=Which set existence axioms are needed to prove the separable Hahn–Banach theorem?|journal=[[Annals of Pure and Applied Logic]]|volume=31|pages=123–144|doi=10.1016/0168-0072(86)90066-7 |doi-access=}} [http://www.math.psu.edu/simpson/papers/hilbert/node7.html#3 Source of citation].</ref><ref>Simpson, Stephen G. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), Cambridge University Press, {{ISBN|978-0-521-88439-6}}, {{MR|2517689}}</ref>