Reference for denseness of testfunctions in sobolevspace

Question

for my thesis I need a reference for a proof that $C_0^\infty(\mathbb R)$ is dense in $W^{2,2}(\mathbb R)$ in respect to the Sobolev-$\| \cdot \|_{W^{2,2}}$-Norm.

I have tried Google but I can't find a proper source to cite from. Do you know some book?

Thanks

Answer 1

There are two theorems you need to cite.

The theorem 1 states that $W^{2,2}(\mathbb R)=W^{2,2}_0(\mathbb R)$, which can be find in page 217, remark 13 in this book Theorem 2 states that $C_0^\infty(I)$ is dense in $W_0^{2,2}(I)$ for any interval $I$, of course for $I=\mathbb R$. You can find this theorem in p211 theorem 8.7 for the version of $W^{1,2}$ in the same book, and read the sentence before section 8.3 on page 217 will finish the argument.

Answer 2

As requested, a reference: Theorem 2.4.1 of lecture notes Sobolev Spaces and Applications by T. Muthukumar.