KKM implies Hahn-Banach

Abstract

Our title means that the Knster-Kuratowski-Mazurkiewitz theorem in1929 implies the Hahn-Banach theorem. This theorem originated fromHahn in 1926 and Banach in 1929 is of basic importance in theanalysis of problems concerning the existence of continuous linearfunctionals. Its consequences and applications cover hundreds ofpapers. For a long period, some authors studied the relation ofresults of the Hahn-Banach theorem and the Brouwer fixed pointtheorem in 1910 (equivalently, the KKM theorem in 1929). In thepresent article, we recall some history of such study, and show thatthe Hahn-Banach theorem can be derived from the KKM theorem and notconversely. Consequently, all consequences and applications of theHahn-Banach theorem belong to a partial realm of the KKM theory.