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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) on some smooth manifold, inside we consider a subspace S⊂C∞(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∞(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.

    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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