Some More Applications of the Hahn-Banach Theorem

Francisco Javier García-Pacheco; Daniele Puglisi; Gusti van Zyl

doi:10.68381/jca21009

Abstract

We give a new proof of the fact that equivalent norms on subspaces can be extended. This new proof is based on the Hahn-Banach Extension Theorem. We also give new characterizations for an equivalent norm on a dual space to be a dual norm. Finally, a new proof of a particular case of the Hahn-Banach Separation Theorem is provided without involving the Axiom of Choice.

Author information

Contact details are reproduced from the original publication and may be historical.

Francisco Javier García-Pacheco

Dept. of Mathematics, University of Cadiz, Puerto Real 11510, Spain

garcia.pacheco@uca.es

Daniele Puglisi

Dept. of Mathematics, University of Catania, Catania 95125, Italy

dpuglisi@dmi.unict.it

Gusti van Zyl

Dept. of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria, South Africa

gusti.vanzyl@up.ac.za

Keywords

Equivalent norm
dual norm
Hahn-Banach
Axiom of Choice

Mathematics Subject Classification

46B20

References

C. Blair: The Baire category theorem implies the principle of dependent choices, Bull. Acad. Pol. Sci., Sr. Sci. Math. Astron. Phys. 25(10) (1977) 933–934.Google Scholar
D. K. Brown, S. G. Simpson: Which set existence axioms are needed to prove the separable Hahn-Banach theorem?, Ann. Pure Appl. Logic 31 (1986) 123–144.https://doi.org/10.1016/0168-0072(86)90066-7 Google Scholar
R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, John Wiley & Sons, New York (1993).Google Scholar
M. Foreman, F. Wehrung: The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set, Fundam. Math. 138 (1991) 13–19.https://doi.org/10.4064/fm-138-1-13-19 Google Scholar
D. Pincus: The strength of Hahn-Banach’s Theorem, in: Victoria Symposium on Non-Standard Analysis, A. Hurd et al. (ed.), Lecture Notes in Math. 369, Springer, Berlin (1974) 203–248.https://doi.org/10.1007/BFb0066014 Google Scholar

Suggested citation

Published by Heldermann Verlag, 2014. Rights now held by Banach Press LLC.

[1] C. Blair: The Baire category theorem implies the principle of dependent choices, Bull. Acad. Pol. Sci., Sr. Sci. Math. Astron. Phys. 25(10) (1977) 933–934.Google Scholar

[2] D. K. Brown, S. G. Simpson: Which set existence axioms are needed to prove the separable Hahn-Banach theorem?, Ann. Pure Appl. Logic 31 (1986) 123–144.https://doi.org/10.1016/0168-0072(86)90066-7 Google Scholar

[3] R. Deville, G. Godefroy, V. Zizler: Smoothness and Renormings in Banach Spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics 64, John Wiley & Sons, New York (1993).Google Scholar

[4] M. Foreman, F. Wehrung: The Hahn-Banach theorem implies the existence of a non-Lebesgue measurable set, Fundam. Math. 138 (1991) 13–19.https://doi.org/10.4064/fm-138-1-13-19 Google Scholar

[5] D. Pincus: The strength of Hahn-Banach’s Theorem, in: Victoria Symposium on Non-Standard Analysis, A. Hurd et al. (ed.), Lecture Notes in Math. 369, Springer, Berlin (1974) 203–248.https://doi.org/10.1007/BFb0066014 Google Scholar