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Evans book on PDE's defines for a given open subset $U$ of $\mathbb{R}^{n}$,

$C^{k}(\overline{U})=\lbrace u:U\rightarrow \mathbb{R}^{n}$, such that $D^{\alpha}u$ exists and is uniformly continuous on bounded subsets of $U$, for all mulitindexes $\alpha$, with $|\alpha|\leq k\rbrace$

Alternatively one could define,

$C^{k}(\overline{U})=\lbrace u:\overline{U}\rightarrow \mathbb{R}^{n},$ such that there exists an open subset $V$ of $\mathbb{R}^{n}$ containing $\overline{U}$, and an extension of $u$ to $V$ that has continous partial derivatives up to order $k$ in $V\rbrace$

Are this definitions equivalent? Is this trivial?

Bill
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  • I'm guessing that maybe we need $\partial U$ to be smooth enough. – Bill Sep 29 '14 at 06:31
  • (first definition just says that the derivatives can be continously extended to $\overline{U}$) – Bill Sep 29 '14 at 06:46
  • Ok, I didn't know about Whitney's extension theorem. Am I correct in assuming this is a direct consequence of it? Are there any conditions to be imposed on $\partial U$? – Bill Sep 29 '14 at 06:56

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