The Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem

Abstract

We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any $L_{\infty }/C_0$ without an uncountable form of the axiom of choice. Moreover, we show that if on each Banach space there exists at least one non-trivial bounded linear functional, then the Hahn-Banach extension theorem must hold. We also discuss relations of non-measurable sets and the Hahn-Banach extension theorem.