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Hi I have the following problem:

Let $X=C([0,1])$ with the $||\cdot||_\infty$-norm. $$A:D(A)\subset X\rightarrow X, Au:=u' , D(A)=C^1([0,1])$$ $$B:D(B)\subset X\rightarrow X, Bu:=u' , D(B)=C^2([0,1])$$ i) Show: $\overline{D(A)}=X$

ii) Is A closed?

iii) Is B closed?

I already showed i) and A is closed. I guess B is not closed but I can't find a proper counterexample. Can someone help me?

Tobi92sr
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  • Have you tried to find a sequence ${ f_n }$ of $C^2$ functions such that $f_n'$ and $f_n$ converge uniformly to $f'$ and $f$, where $f$ is a $C^1$ function that is not in $C^2$? – Disintegrating By Parts Nov 24 '17 at 16:36

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