In the book "Distributions- theory and applications" by Duistermaat & Kolk the authors use the following lemma (Lemma 11.10 on page 123) in a combination with the Hahn-Banach theorem to prove that the tensor product of test functions (i.e $\varphi \otimes \psi, \varphi \in C_0^{\infty}(X), \psi \in C_0^{\infty}(Y)$) is dense in the space $C_0^{\infty}(X \times Y)$:
Let X and Y be open subsets of $\mathbb{R}^n$ and $\mathbb{R}^p$ respectively, and $\omega \in \mathscr{D'}(X \times Y)$. If $\omega(\varphi \otimes \psi)=0$ for all $\varphi \in C_0^{\infty}(X), \psi \in C_0^{\infty}(Y)$, then $\omega=0$.
I have some questions on the proof of this proposition
Proof: Let $(x_0,y_0) \in X \times Y$. We can find a corresponding neighbourhood $U$ of $(x_0,y_0)$ whose closure is a compact subset $K$ of $X \times Y$.If $\chi \in C_0^{\infty}(X \times Y)$ equals one on an open neighbourhood of $K$, one has $\chi \omega \in \mathcal{E'}(X \times Y)\subset \mathscr{D'}(X \times Y)$. Then we get, if $\epsilon >0$ is sufficiently small:
$(\chi \omega ) \ast (\varphi_{\epsilon} \otimes \psi_{\epsilon})(x,y)= (\chi \omega) (T_x S \varphi_{\epsilon} \otimes T_y S \psi_{\epsilon} )= \omega(T_x S \varphi_{\epsilon} \otimes T_y S \psi_{\epsilon} )$ equals 0 for all $(x,y) \in K$, in view of $\text{supp} T_z S \zeta_{\epsilon}=z + (- \epsilon \ \text{supp} \zeta)$. On account of Lemma 11.6, the limit of the left-hand side , as $\epsilon \to 0$, equals $\chi \omega$; thus we obtain $\chi \omega =0 $ on $U$, and so $\omega =0$ on $U$.
(They refer to Lemma 11.6 which states that if $\varphi \in C_0^{\infty}(\mathbb{R}^n)$ such that $1(\varphi)=1$, $u \in \mathscr{D'}(\mathbb{R}^n)$, then $u \ast \phi_{\epsilon} \in C^{\infty}(\mathbb{R^n}) $ and $u \ast \varphi_{\epsilon} \to u$ as $\epsilon \to 0$ (where $\varphi_{\epsilon}(x)= \frac{1}{\epsilon^n}\varphi(\frac{x}{\epsilon})$).)
My questions are these:
$\cdot$ If I understand it correctly $\chi \omega \in \mathcal{E'}(X \times Y)$ implies that $(\chi \omega ) \ast (\varphi_{\epsilon} \otimes \psi_{\epsilon})(x,y) \in C_0^{\infty}(X \times Y)$, but what's the point in doing this?
$\cdot$ They write "$ \omega(T_x S \varphi_{\epsilon} \otimes T_y S \psi_{\epsilon} )$ equals 0 for all $(x,y) \in K$, in view of $\text{supp} T_z S \zeta_{\epsilon}=z + (- \epsilon \ \text{supp} \zeta)$." Why is this? Doesn't it follow that $\omega(T_x S \varphi_{\epsilon} \otimes T_y S \psi_{\epsilon} )$ from the assumption that $\omega(\varphi \otimes \psi)=0$ for all $\varphi \in C_0^{\infty}(X), \psi \in C_0^{\infty}(Y)$?