Linear extenders and the Axiom of Choice

Abstract

In set theory without the axiom of Choice ZF, we prove that for every commutative field IK, the following statement D_{\IK}: "On every non null IK-vector space, there exists a non null linear form" implies the existence of a IK-linear extender on every vector subspace of a $\IK$-vector space. This solves a question raised in \cite{Mo09}. In the second part of the paper, we generalize our results in the case of spherically complete ultrametric valued fields, and show that Ingleton's statement is equivalent to the existence of "isometric linear extenders".