KKM implies Hahn-Banach

Abstract

Our title means that the Knaster-Kuratowski-Mazurkiewicz theorem in 1929 implies the Hahn-Banach theorem. This theorem originated from Hahn in 1926 and Banach in 1929 is of basic importance in the analysis of problems concerning the existence of continuous linear functionals. Its consequences and applications cover hundreds of papers. For a long period, some authors studied the relation of results of the Hahn Banach theorem and the Brouwer FIxed point theorem in 1912 (equivalently, the KKM theorem in 1929). In the present article, we recall some history of such study, and show that the Hahn-Banach theorem can be derived from the KKM theorem and not conversely. Consequently, all consequences and applications of the Hahn-Banach theorem belong to a partial realm of the KKM theory.