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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 11th 2013
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeAug 10th 2017

Is there a form of the Whitney extension theorem in the generality of Fréchet manifolds?

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeAug 10th 2017

I did look into this, I’d have to go back and check. It depends on what sort of closed sets you allow the initial functions on. If you have a mapping space it might be easier.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeAug 11th 2017
• (edited Aug 11th 2017)

Thanks, David. Yes, I’d just need it for mapping spaces (or spaces of sections). Also, I don’t need any constraints on the derivatives, just the statement that a smooth function has an extension. Thinking about it though, I realize that I don’t need it for compact subspaces, but for closed subspaces. Hm, maybe I am not asking an optimal question here.

What I am really wondering is what to make of definition 19 in Collini 16. Here the ambient Fréchet space is that of smooth functions $C^\infty(X)$ on some smooth manifold, inside we consider a subspace $S \subset C^\infty(X)$ of those that satisfy some differential equation (a Klein-Gordon equation with interaction inhomogenity).

Collini wants to talk about smooth functions on the Fréchet submanifold $S$. In definition 19 he defines these to be those functions on $S$ for which there exists an extension to a smooth function on a neighbourhood of $S$ in $C^\infty(X)$. (He also has constraints on wave front sets of their functional derivatives, but I think this is an issue to be dealt with separately, so let’s ignore this).

All I am wondering is whether this defintion 19 (without the wave front condition) secretly reduces simply to smooth functions on $S$.

• CommentRowNumber5.
• CommentAuthorDavidRoberts
• CommentTimeMar 4th 2019

Added actual end statement of Roberts–Schmeding, namely that in the nonlinear mapping space case one gets a submersion of Fréchet manifolds.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMar 5th 2019

Thanks, David. But this shouldn’t really be discussed in the References-section. Best to state your theorem as a theorem in the main part of the entry!

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeMar 5th 2019

OK, will do!