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nLab > Latest Changes: Whitney extension theorem

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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 ∞(X)C^\infty(X) on some smooth manifold, inside we consider a subspace S⊂C ∞(X)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 SS. In definition 19 he defines these to be those functions on SS for which there exists an extension to a smooth function on a neighbourhood of SS in C ∞(X)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 SS.

    • 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.

    diff, v7, current

    • 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!

    • CommentRowNumber8.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 25th 2019

    Added Taimanov theorem to related entries

    diff, v9, current

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