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Seemingly completely forgotten, back in 1936, the Dutch mathematician Freudenthal, quite well known at the time, proved his so called Spectral Theorem, see chapter 6 in Luxemburg & Zaanen : Riesz Spaces I. North-Holland, 1971. The amusing thing is that the theorem is formulated eclusively in terms of partial orders, and on top of it, its proof is also in the very same terms. Yet one of the rather direct consequences of it is the spectral representation of normal operators in Hilbert spaces. Another one is the Radon-Nykodim theorem in measure theory. And to aggravate things, it can also solve some Poisson PDEs. Does anybody know about more recent applications of that theorem ? And how about having more appreciation for the concept of partial order ?

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Just a small clarification about Hans Freudenthal. He was born into a German Jewish family in 1905 and studied in Berlin for his doctorate with Heinz Hopf. Then he went to Amsterdam as assistant to Brouwer, where he had trouble surviving the war years. After the war he became a professor at Utrecht. He did spend most of his long life in the Netherlands and influenced many students there. – Jim Humphreys Jul 13 '10 at 10:39
Thank you for the biographical details which may be instructive to those younger among us ... – Elemer E Rosinger Jul 13 '10 at 15:31
Could you sketch the statement of the result, or provide a pointer to somewhere online? – András Salamon Aug 1 '10 at 21:59
Is there a pure partial order version? The version I know requires a vector lattice. – arsmath Oct 29 '15 at 13:04

I don't know if this can be considered "recent", but there is a well-known application to some demonstrations around spheres homotopy, see for example

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I am very doubtful that this has anything to do with the Freudenthal theorem mentioned in the question... – Yemon Choi Jul 13 '10 at 9:22
Sorry, the website mentioned above does not function ... – Elemer E Rosinger Jul 13 '10 at 15:33
The point is that if you Google for "Freudenthal suspension theorem" you do get a result due to Freudenthal about higher homotopy groups of spheres. I would be amazed if this had any connection to the 1936 spectral theorem you describe. See – Yemon Choi Jul 13 '10 at 18:07

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