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

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.

• To get #1 from #2 and #3, you can use the general topological fact: if we have $X,Y,Z$ subspaces of some fixed topological space $A$, with $X \subset Y \subset Z$, and we have $\overline{X} \supset Y$ and $\overline{Y} \supset Z$, then $\overline{X} \supset Z$. This follows from the fact that $\overline{\overline{X}}=\overline{X}$, i.e. the closure operation is idempotent. – Ian Aug 26 '15 at 18:12
• @Ian, thanks. Fact #1 results from fact #2 and the fact that $C_0^\infty \subset W_2^2 \subset L^2$, right? – Glinka Aug 26 '15 at 18:19
• Err...I misspoke slightly. Fact #2 arises from fact #3 and fact #1, through the argument I gave. – Ian Aug 26 '15 at 18:21
• @Ian, well, we can show #1 from #2, or #2 from #1 and #3. The third fact appears to be the strongest. Anyway, the proper literature is what I'm after. – Glinka Aug 26 '15 at 18:30
• Well, I studied out of Evans, which is pretty standard, at least if your objective is to apply these techniques to PDEs. – Ian Aug 26 '15 at 18:37

## 1 Answer

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.

• I read the French version of Brezis' book many years ago, and I remember he focuses (like many other books with a focus on PDE) mostly on $W^{1,p}$ (he proves analogous results in that case). Now, I don't have a paper copy at hand, I am not sure the English version is identical to the French one, and most of all I don't trust my memory. Am I wrong? Thanks. – Silvia Ghinassi Aug 26 '15 at 18:30
• I briefly looked into Haim Brezis book here: fmi.uni-sofia.bg/lecturers/ma/milen/Brezis_FA_PDE.pdf, but i did find anything about bump functions or its density in Sobolev space :( – Glinka Aug 26 '15 at 18:41
• @Dennis Welcome to the world of nonlinear analysis! You never find the exact statement that you need ;-) Actually, the density of compactly supported functions is proved in both books for the space $W^{1,p}$. The general case is true although it is not precisely stated as a theorem. Another possible source is Giovanni Leoni's book on Sobolev spaces, published by the American Mathematical Society. – Siminore Aug 26 '15 at 19:55