Linear extenders and the Axiom of Choice

Morillon, Marianne. "Linear extenders and the Axiom of Choice." Commentationes Mathematicae Universitatis Carolinae 58.4 (2017): 419-434. <http://eudml.org/doc/294435>.

@article{Morillon2017,
abstract = {In set theory without the Axiom of Choice ZF, we prove that for every commutative field $\mathbb \{K\}$, the following statement $\mathbf \{D\}_\{\mathbb \{K\}\}$: “On every non null $\mathbb \{K\}$-vector space, there exists a non null linear form” implies the existence of a “$\mathbb \{K\}$-linear extender” on every vector subspace of a $\mathbb \{K\}$-vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. 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”.},
author = {Morillon, Marianne},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Axiom of Choice; extension of linear forms; non-Archimedean fields; Ingleton's theorem},
language = {eng},
number = {4},
pages = {419-434},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Linear extenders and the Axiom of Choice},
url = {http://eudml.org/doc/294435},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Morillon, Marianne
TI - Linear extenders and the Axiom of Choice
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 4
SP - 419
EP - 434
AB - In set theory without the Axiom of Choice ZF, we prove that for every commutative field $\mathbb {K}$, the following statement $\mathbf {D}_{\mathbb {K}}$: “On every non null $\mathbb {K}$-vector space, there exists a non null linear form” implies the existence of a “$\mathbb {K}$-linear extender” on every vector subspace of a $\mathbb {K}$-vector space. This solves a question raised in Morillon M., Linear forms and axioms of choice, Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. 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”.
LA - eng
KW - Axiom of Choice; extension of linear forms; non-Archimedean fields; Ingleton's theorem
UR - http://eudml.org/doc/294435
ER -