# Basic question on Sobolev spaces

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.

• Wouldn't you have even $H^1(0,1)\subset C[0,1]$? – timur Sep 27 '12 at 11:37
• Yes, due to the Sobolev Embedding Theorem. – Dina Sep 28 '12 at 11:24
• Would you even have $W^{1,1}(0,1)\subset C[0,1]$? – timur Sep 28 '12 at 14:29

## 1 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.

• Thank you, that was really easy! – Dina Sep 27 '12 at 11:28