Talk:Differentiability class
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The formulations are not clear. Since R4 has exotic smooth structures, it is not right as posed. And it is not true that the Ck are all the same. The intended meanings should be clarified. Charles Matthews 22:36, 7 October 2005 (UTC)
Moved this from the page:
In the case of a Lie group, it turns out that the sets of Ck are essentially the same for all k ≥ 0 (including ∞ and ω). Much like the case of the possible differentiability classes of complex functions, only stronger, since it also extends to the k=0 case, this result shows that Lie groups admit essentially only one differential structure. This is the content of Hilbert's fifth problem, first posed by Lie, and proved by Gleason, Montgomery, and Zippin in the 1950s. A related proposition, considered by some to be a better formulation of Hilbert's fifth problem, is the Hilbert-Smith conjecture, an open problem.
Obviously on Euclidean space, which is a Lie group, continuous functions and smooth functions differ. So this is wrong as it stands. I think a reference to the Hilbert 5 page, which I have stopped redirecting to Lie group, is better than trying to say here what this ought to say. Charles Matthews 11:18, 30 October 2005 (UTC)
