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I am quite new to the field of Sobolev spaces. So, I want to apologize in advance, if the question is too obvious!

I have a problem with understanding the connection between the Hilbert space $H^2(0,1)$ and $C^2[0,1]$. I know, that due to the Sobolev Embedding theorem the embedding $E: H^2(0,1)\to C[0,1] (\text{and} C^1[0,1])$ is compact. But what could be said about $C^2[0,1]$? Can we say, that $C^2[0,1]\subset H^2(0,1)$?

I would really appreciate the answer.

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  • $\begingroup$ Wouldn't you have even $H^1(0,1)\subset C[0,1]$? $\endgroup$
    – timur
    Sep 27, 2012 at 11:37
  • $\begingroup$ Yes, due to the Sobolev Embedding Theorem. $\endgroup$
    – Dina
    Sep 28, 2012 at 11:24
  • $\begingroup$ Would you even have $W^{1,1}(0,1)\subset C[0,1]$? $\endgroup$
    – timur
    Sep 28, 2012 at 14:29

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Yes, because you have a bound $$\int_0^1\lvert f^{(k)}(x)\rvert^2\,dx\le \lVert f^{(k)}\rVert_\infty^2\qquad\text{for }k=0,1,2,$$ thus bounding the $H^2$ norm of $f$ in terms of the $C^2$ norm. Here $\lVert\cdot\rVert_\infty$ is the sup norm.

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  • $\begingroup$ Thank you, that was really easy! $\endgroup$
    – Dina
    Sep 27, 2012 at 11:28

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