The Hahn-Banach theorem: the life and times

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 1900s. Hahn and Banach independently proved the theorem for the real case in the 1920s. Then there was Murrayʹs extension to the complex case—easy, once you realize that (χ) = Re (χ) − iRe (iχ). Can continuous linear maps 06 be extended as easily as linear functionals? Banach and Mazur had already proved that they could not in 1933 but it was not until Nachbinʹs 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.