Talk:Peetre theorem

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[edit] Continuity hypotheses

Judging from Peetre's paper, the only necessary continuity hypothesis is that D be a morphism of sheaves. I find this a little bit unbelievable, and I would think D should be continuous under the stronger Fréchet-Schwarz topology. The Terng paper explicitly imposes such a condition. Rather than agonize over what the correct continuity condition is, I have settled for the weaker condition. But take this with a grain of salt. Cheers, Silly rabbit 15:51, 12 November 2005 (UTC)

Okay, I've convinced myself that it is true under the weaker hypothesis. I can more-or-less prove it with these hypotheses, although my "proof" is omitted because it adds very little to the understanding of the theorem. Interested parties should contact me here or on my talk page. Regards, Silly rabbit 05:44, 15 November 2005 (UTC)

I added the proof. In my opinion this result is counter-intuitive enough to require it. The proof still needs some work, though. Silly rabbit 09:35, 17 November 2005 (UTC)

Ok, the proof is complete with (almost all) bells and whistles attached. I realize it's very un-wikipedian to include proofs when they don't actually add to an understanding of the theorem. However, Peetre's original proof is clearly unsatisfactory. Furthermore, his erratum is overly general in my opinion since it uses distributions instead of sections. (The really fascinating direction is how this theorem generalizes for non-linear operators, but any such generalization would certainly be too complicated to state here.) I believe that the theorem so defies a mathematician's instincts against falsehood that a detailed proof absolutely must be presented here. I posed it as an exercise for a very bright colleague of mine who works in hard analysis, or -- if there is such a thing -- very hard analysis. In particular, he is currently working on a massive three-part paper to generalize the Whitney extension theorem to jets. His initial reaction was that the theorem can't possibly be true. After letting him steep for awhile, I fed him the broad strokes of the proof. Anyway, partly due to his initial reaction, I felt that it was best to include the proof here. Regards, Silly rabbit 21:58, 18 November 2005 (UTC)