Density of spaces $C_0^{\infty}(\mathbb{R})$, $W_2^2(\mathbb{R})$ and $L^2(\mathbb{R})$ in each other

Question

Let's consider following spaces:

• $L^2(\mathbb{R}) = L^2(\mathbb{R}, \mathbb{C}, \mu_L)$ --- space of $\mathbb{C}$-valued functions defined on $\mathbb{R}$ for which the square of the absolute value is Lebesgue integrable relatively to standart Lebesgue measure. This space is considered as quotient by subspace of functions that equals zero almost everywhere.

• $W_2^2(\mathbb{R}) = \{f\in L^2(\mathbb{R}):\,\,\exists f',f''\in L^2(\mathbb{R})\}$ --- Sobolev space, subspace of $L^2(\mathbb{R})$, consists of functions that are twice differentiable in a week sense with derivatives in $L^2(\mathbb{R})$.

• $C_0^{\infty}(\mathbb{R}) = \{f\in C^{\infty}(\mathbb{R}):\,\,\overline{\mathrm{supp}\,f}\subsetneq \mathbb{R}\}$ --- space of bump functions, i.e. functions that are both smooth, in the sense of having continuous (strong) derivatives of all orders, and compactly supported. This space is considered as a subspaces of $L^2(\mathbb{R})$.

The statements are following:

1. $W_2^2(\mathbb{R})$ is dense in $L^2(\mathbb{R})$, i.e. for every $f\in L^2$ there exist a sequence $\{f_n \in W_2^2(\mathbb{R})\}_{n=1}^{\infty}$ such that $\lim\limits_{n\to \infty}\|f-f_n\|_{L^2} = 0$;
2. $C_0^{\infty}(\mathbb{R})$ is dense in $L^2(\mathbb{R})$;
3. $C_0^{\infty}(\mathbb{R})$ is dense in $W_2^2(\mathbb{R})$ relatively to $W_2^2$ metric, i.e. for every $v\in W_2^2(\mathbb{R})$ there exist a sequence $\{v_n \in C_0^{\infty}(\mathbb{R})\}_{n=1}^{\infty}$ such that $\lim\limits_{n\to \infty}\|v-v_n\|_{W_2^2} = 0$, which means that $\lim\limits_{n\to \infty}\|v-v_n\|_{L^2} = 0$, $\lim\limits_{n\to \infty}\|v'-v_n'\|_{L^2} = 0$ and $\lim\limits_{n\to \infty}\|v''-v_n''\|_{L^2} = 0$.

The question is where can I look these facts up? I'm looking for the most classical and canonical source possible.

Answer 1

The second and the third statement are basic facts in Sobolev Space theory. The first one is probably a corollary of the other two, but I think it can be proved directly. Anyway, Michel Willem's book on Functional Analysis, Birkhauser-Verlag, contains the proofs of these facts. You can also read Haim Brezis's book on functional analysis (Springer-Verlag) for a more detailed presentation.

I personally use these textbooks for my lectures on Sobolev spaces and applications to PDEs.