"I love the Hahn-Banach Theorem"
No, not me. I'm just quoting:
Thus begins the abstract to the eulogy "On the Hahn-Banach Theorem" by Lawrence Narici.
I love the Hahn-Banach theorem. I love it the way I love Casablanca and the Fontana di Trevi. It is something not so much to be read as fondled.
Thus begins the abstract to the eulogy "On the Hahn-Banach Theorem" by Lawrence Narici.
There's not much I can add to that. Have you ever felt this way about a theorem?
(I have a virulent dislike of analysis but I'm going to see if this paper can convert me.)
Labels: mathematics
posted by sigfpe at Monday, July 24, 2006
2 Comments:
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.)
The Hahn-Banach theorem is fundamental in the sense that it states that the dual of a locally convex vector space is non zero.
It is the most useful form of the axiom of choice in the category of topological vector spaces.
It actually implies (is equivalent to ?) the "axiom of choice for the category of topological v. spaces", ie. each epic has a section.
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.
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