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.
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.