Without the Hahn-Banach theorem, functional analysis would be very different from the structure we know today. Among other things, it has proved to be a very appropriate form of the Axiom of Choice for the analyst. (It is not equivalent to the Axiom of Choice, incidentally; it follows from the ultrafilter theorem which is strictly weaker.) Riesz and Helly obtained forerunners of the theorem in the turbulent mathematical world of the early 1900's. Hahn and Banach independently proved the theorem for the real case in the 1920's. Then there was Murray's extension to the complex case - easy, once you realize that f(x) = Re f(x) - i Re f(ix). Can continuous linear maps be extended as easily as linear functionals? Banach and Mazur had already proved that they could not in 1933 but it was not until Nachbin's 1950 result that a definitive answer was achieved to this more general question. In this article, we discuss the mathematical world into which the theorem entered, examine its connection to the axiom of choice, look at some ancestors, mention some of its consequences and consider some of its principal variations.
Reference: Topology Appl. 77 (1997) no. 2, 193-211
Published version: http://www.sciencedirect.com/science/article/B6V1K-3WNMXF9-D/1/9b2bd45b3bf0f9e7e69534c03fdf70ee
Date received: February 22, 1996.
Date published: March 24, 1996.
Copyright © 1996 by the authors. Distributed by Topology Atlas with permission of the authors. Production Editor: K.P. Hart.