On the Equivalence in ZF+BPI of the Hahn–Banach Theorem and Three Classical Theorems

The presented paper is a compendium or a kind of précis of relationships between the four classical theorems of mathematical analysis. More precisely, the first aim of this paper is to present the equivalence of the four classical theorems: the Hahn–Banach theorem, the Mazur–Orlicz theorem, the Markov–Kakutani fixed-point theorem and the von Neumann theorem on amenability of Abelian groups. The second purpose is to prove these equivalences in Zermelo–Fraenkel set theory with the axiom of choice replaced by the Boolean prime ideal theorem, from which we can also prove the Hahn–Banach theorem.