I am looking at the following theorem:
- $C_C^{\infty}(\mathbb{R}^n) \subset S(\mathbb{R}^n)$ and the embedding is continuous.
- $C_C^{\infty}(\mathbb{R}^n)$ is dense in $S(\mathbb{R}^n)$
- $S(\mathbb{R}^n) \subset L_1(\mathbb{R}^n)$ and the embedding is continuous.
Note that a function $\phi \in C^{\infty}(\mathbb{R}^n)$ for which $||\phi||_{\alpha, \beta}=\sup_x |x^{\alpha} D^{\beta} \phi|<+\infty \ \ \ \forall \alpha, \beta$ belongs to $S(\mathbb{R}^n)$.
Let $\phi \in C_C^{\infty}(\mathbb{R}^n)$. We have that $supp (D^{\beta} \phi) \subset supp(\phi)$ since $D^{\beta} \phi \neq 0 \Rightarrow \phi \neq 0$, right? From this we have that $D^{\beta}\phi \in C_C^{\infty}(\mathbb{R}^n)$, thus also $x^{\alpha} D^{\beta} \phi \in C_C^{\infty}(\mathbb{R}^n), \forall \alpha$. And so we deduce that $\sup_x |x^{\alpha} D^{\beta} \phi|<+\infty \Rightarrow \phi \in S(\mathbb{R}^n)$.
Right? But how can we show that the embedding is continuous?
Let $\phi \in S(\mathbb{R}^n)$. Let $\rho(x)$ such that $\int_{\mathbb{R}^n} \rho(x) dx=1, \ \ \ supp{\rho } \subset \{ ||x|| \leq 1\}$ . Consider $\phi_j(x)=\rho{\left( \frac{x}{j}\right)} \phi(x) \in C_C^{\infty}(\mathbb{R}^n)$.
So it suffices to show that $\phi_j(x) \to\phi(x)$. How could we show this?
- Let $\phi \in S(\mathbb{R}^n)$. Then $ \sup_x |x^{\alpha} D^{\beta}\phi |<+\infty$ . For $\alpha=0, \beta=0$ we get that $\sup_x|\phi|<+\infty$. So $\int |\phi(x)| dx<+\infty$ and so $\phi \in L_1(\mathbb{R}^n)$. Is this right? How can we show that the embedding is continuous?