The equivalency of the Banach-Alaoglu theorem and the axiom of choice - Kumar, Nand Kishor; Raikhola, Sher Singh (Cognition, 2024)
The Axiom of Choice can be used to prove the Banach-Alaoglu theorem, while a weakened form of the Axiom of Choice is required to prove the full strength of the Hahn-Banach theorem, which is equivalent to the Banach-Alaoglu theorem. Although the Banach-Alaoglu theorem and the full- strength Axiom of Choice are not exactly equivalent, they are closely related. The Banach-Alaoglu theorem is a compactness theorem whose proof mainly depends on Tychonoff’s theorem. In this article, Banach-Alaoglu theorem is equivalent to the axiom of choice, has been proved.