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 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