I just had a quick question about a notation I saw. I know that a $C^2$ function is a function whose second partial derivative exists. But, what does it mean when one writes $C^2(U)$ where $U$ is a subset of $R^n$? Thanks in advance.
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It means that $f$ is $C^2$ on $U$. It is just to specify to a domain on which we have the $C^2$ property.
If $U$ happens to be non-open, then it means that there exists $\widetilde{U}$ an open neighborhood of $U$ and $F\in C^2(\widetilde{U})$ such that $F_{\vert U}=f$.
C. Falcon
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What is that notation with the |U you used there? – Jan 01 '18 at 23:49
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You definitely hate notation, just joking! (: It stands for the restriction of $F$ at $U$. – C. Falcon Jan 01 '18 at 23:59