# Article

Full entry | PDF   (0.3 MB)
Keywords:
Axiom of Choice; extension of linear forms; non-Archimedean fields; Ingleton's theorem
Summary:
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''.
References:
[1] Blass A.: Existence of bases implies the axiom of choice. in Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math., 31, Amer. Math. Soc., Providence, RI, 1984, pp. 31–33. DOI 10.1090/conm/031/763890 | MR 0763890 | Zbl 0557.03030
[2] Bleicher M.N.: Some theorems on vector spaces and the axiom of choice. Fund. Math. 54 (1964), 95–107. DOI 10.4064/fm-54-1-95-107 | MR 0164899 | Zbl 0118.25503
[3] Hodges W.: Model Theory. Encyclopedia of Mathematics and its Applications, 42, Cambridge University Press, Cambridge, 1993. MR 1221741 | Zbl 1139.03021
[4] Howard P., Rubin J.E.: Consequences of the Axiom of Choice. Mathematical Surveys and Monographs, 59, American Mathematical Society, Providence, RI, 1998. DOI 10.1090/surv/059 | MR 1637107 | Zbl 0947.03001
[5] Howard P., Tachtsis E.: On vector spaces over specific fields without choice. MLQ Math. Log. Q. 59 (2013), no. 3, 128–146. DOI 10.1002/malq.201200049 | MR 3066735 | Zbl 1278.03082
[6] Ingleton A.W.: The Hahn-Banach theorem for non-Archimedean valued fields. Proc. Cambridge Philos. Soc. 48 (1952), 41–45. MR 0045939 | Zbl 0046.12001
[7] Jech T.J.: The Axiom of Choice. North-Holland Publishing Co., Amsterdam, 1973. MR 0396271 | Zbl 0259.02052
[8] Luxemburg W.A.J.: Reduced powers of the real number system and equivalents of the Hahn-Banach extension theorem. in Applications of Model Theory to Algebra, Analysis, and Probability (Internat. Sympos., Pasadena, Calif., 1967), Holt, Rinehart and Winston, New York, 1969, pp. 123–137. MR 0237327 | Zbl 0181.40101
[9] Morillon M.: Linear forms and axioms of choice. Comment. Math. Univ. Carolin. 50 (2009), no. 3, 421-431. MR 2573415 | Zbl 1212.03034
[10] Narici L., Beckenstein E., Bachman G.: Functional Analysis and Valuation Theory. Pure and Applied Mathematics, 5, Marcel Dekker, Inc., New York, 1971. MR 0361697 | Zbl 0218.46004
[11] Schneider P.: Nonarchimedean Functional Analysis. Springer Monographs in Mathematics, Springer, Berlin, 2002. MR 1869547 | Zbl 0998.46044
[12] van Rooij A.C.M.: Non-Archimedean Functional Analysis. Monographs and Textbooks in Pure and Applied Math., 51, Marcel Dekker, Inc., New York, 1978. MR 0512894 | Zbl 0396.46061
[13] van Rooij A.C.M.: The axiom of choice in $p$-adic functional analysis. In $p$-adic functional analysis (Laredo, 1990), Lecture Notes in Pure and Appl. Math., 137, Dekker, New York, 1992, pp. 151–156. MR 1152576 | Zbl 0781.46055
[14] Warner S.: Topological Fields. North-Holland Mathematics Studies, 157; Notas de Matemática [Mathematical Notes], 126; North-Holland Publishing Co., Amsterdam, 1989. MR 1002951 | Zbl 0683.12014

Partner of