tag:blogger.com,1999:blog-11295132.post115375753503072105..comments2014-04-16T10:57:46.206-07:00Comments on A Neighborhood of Infinity: "I love the Hahn-Banach Theorem"Dan Piponihttps://plus.google.com/107913314994758123748noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-11295132.post-418474119977722082011-04-17T05:42:30.321-07:002011-04-17T05:42:30.321-07:00The Hahn-Banach theorem is fundamental in the sens...The Hahn-Banach theorem is fundamental in the sense that it states that the dual of a locally convex vector space is non zero.<br /><br />It is the most useful form of the axiom of choice in the category of topological vector spaces.<br /><br />It actually implies (is equivalent to ?) the "axiom of choice for the category of topological v. spaces", ie. each epic has a section.<br /><br />The fact that H-B has a proof in the separable Hilbert case without AC implies that many properties of separable Hilbert spaces have nothing to do with AC.kebabhttp://www.blogger.com/profile/02175738068903229398noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1154121906648312292006-07-28T14:25:00.000-07:002006-07-28T14:25:00.000-07:00I'm sure there are things in analysis that some pe...I'm sure there are things in analysis that some people find prettier than the Hahn-Banach theorem. But analysis is probably my weakest branch of mathematics, despite doing my PhD in Riemann surface stuff which is largely complex analysis. As a result, the Hahn-Banach theorem is about as far as I ever got in the subject. And even then I don't find the usual statement of it terribly intuitive. (I suspect an analyst would have trouble understanding why I could possibly fail to get it.)sigfpehttp://www.blogger.com/profile/08096190433222340957noreply@blogger.comtag:blogger.com,1999:blog-11295132.post-1153882058792364812006-07-25T19:47:00.000-07:002006-07-25T19:47:00.000-07:00The Hahn-Banach theorem is not a good reason to lo...The Hahn-Banach theorem is not a good reason to love analysis, and there are far more beautiful things in analysis than the stuff in this paper.GeometricAnalysthttp://www.blogger.com/profile/12352215420460864176noreply@blogger.com