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

### Martin Väth

Czech Academy of Sciences, Prague, Czech Republic### W.A.J. Luxemburg

California Institute of Technology, Pasadena, USA

## Abstract

We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any $L_{∞}/C_{0}$ 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.

## Cite this article

Martin Väth, W.A.J. Luxemburg, The Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem. Z. Anal. Anwend. 20 (2001), no. 2, pp. 267–279

DOI 10.4171/ZAA/1015