let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set.

In Evans book we find the definition

$$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on bounded subsets of} U, \text{ for all }|\alpha|\leq k \}$$

it is easy to show that this is equivalent to

$$ C^k(\overline U) = \{f\in C^k(U)| D^\alpha f \text{ can be extended continuously on } \overline U, \text{ for all } \}$$

In differential geometry the definition

$$ C^k(\overline U) = \{f|_{\overline U} \colon \exists O\supset \overline U \text{ such that } f\in C^k(O)\}$$

Is it obvious that the first two definitions are equivalent to the last one?

Edit: Or was this identity first shown in Whitney's extension theorem?

Edit 2: Conisdering Guiseppe Negro's comment: Since the last definition is used in differential geometry, is it true that those definitions are equivalent when the set $\overline U$ is a manifold or manifold with boundary?

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    $\begingroup$ They are not equivalent, the second condition being stronger. If a domain is not sufficiently smooth, a function defined in it can be smooth without being the restriction of a global smooth function. This can be a major nuisance, especially (AFAIK) in the theory of Sobolev spaces. $\endgroup$ Commented Mar 23, 2014 at 21:02
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    $\begingroup$ I am sorry that I cannot contribute more to this and the other thread (the one with the bounty on it). Anyway, the standard example of a domain for which the second definition yields a strictly smaller space is due to Lebesgue and it is the two-dimensional domain $$ \{-1\le x \le 1,\ -1\le y \le 1\ :\ \lvert y \rvert \ge \begin{cases} e^{-1\x}, & x>0 \\ 0, x \le 0 \end{cases}\}.$$(It is a square with an entering sharp cusp). The function $$f(x,y)=\begin{cases}x^2 & x> 0,\ y>0 \\ 0 & \text{otherwise}\end{cases}$$ is $C^1$ in the interior of this domain [...] $\endgroup$ Commented Apr 3, 2014 at 11:12
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    $\begingroup$ [...] and can also be extended to the boundary together with its first derivative. However, it cannot be the restriction of any global $C^1$ function, because such a function wouldn't be Lipschitz continuous in a neighborhood of the origin. $\endgroup$ Commented Apr 3, 2014 at 11:16
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    $\begingroup$ Hope this helps. Anyway, the standard resource for questions of this kind in the category of Sobolev spaces is AFAIK the book by Adams. $\endgroup$ Commented Apr 3, 2014 at 11:18
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    $\begingroup$ Regarding edit 2: Yes, those definitions are equivalent if you have a smooth enough boundary, for in that case you can construct an extension operator by performing a reflection locally at any point of the boundary. This is surely explained in Adams's book, in the chapter on "extension theorems". Constructing an extension operator for $C^k$ functions is the first step towards the construction of an extension operator for Sobolev functions. $\endgroup$ Commented Apr 3, 2014 at 11:30


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