The Hahn-Banach theorem: the life and times

✍ Scribed by Lawrence Narici; Edward Beckenstein

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 765 KB
• Volume: 77
• Category: Article
• ISSN: 0166-8641
• DOI: 10.1016/s0166-8641(96)00142-3

✦ Synopsis

Without the Hahn-Banach theorem, functional analysis would he ve.,aj different from the strncture 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 ultrafiher theorem witich is strictly weaker.) Riesz and Heliy oblained forerunnc~ of the theorem in the turbulent mathematical wodd of the early 1900s. Hahn and Bannch independently proved the theorem for the real case in the 1920s. Then there was Murray's extension m the complex case--easy, once you realize that f(x) = Re f(a:) -iRe f(iz). Can continuous linear maps he extended as easily as linear fuuctionals? Banash and Mazur had already proved that they could not in 1933 but it was not until Nachbin's 1950 result that a definitive answer w~ achieved to this more general question, in this asticle, we discuss the mathematical world into which the theorem entered, examine its connection to the axiom of choice, look m some ancestl~, ment~ou some of its consequences and consider soma of its principal variations.