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Zeitschrift für Analysis und ihre Anwendungen


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Volume 20, Issue 2, 2001, pp. 267–279
DOI: 10.4171/ZAA/1015

Published online: 2001-06-30

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

W.A.J. Luxemburg[1] and Martin Väth[2]

(1) California Institute of Technology, Pasadena, USA
(2) Czech Academy of Sciences, Prague, Czech Republic

We show that it is impossible to prove the existence of a linear (bounded or unbounded) functional on any $_\{infty}/C_0$ 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.

Keywords: Power of the Hahn-Banach theorem, linear functionals, axiom of choice, axiom of dependent choices, Shelah’s model

Luxemburg W.A.J., Väth Martin: The Existence of Non-Trivial Bounded Functionals Implies the Hahn-Banach Extension Theorem. Z. Anal. Anwend. 20 (2001), 267-279. doi: 10.4171/ZAA/1015