Proposition 4

a) The canonical injection of $\mathscr{D}(U)$ into $\mathscr{K}(U)$ is continuous and $\mathscr{D}(U)$ is dense in $\mathscr{K}(U)$;

b) For every $p\in [1,+\infty [$, the canonical injection of $\mathscr{D}(U)$ into $\mathscr{L}^p(U)$ is continuous and $\mathscr{D}(U)$ is dense in $L^p(U)$.

Every continuous seminorm on $\mathscr{K}(U)$ is continuous on $\mathscr{D}$(U), hence the canonical injection of $\mathscr{D}(U)$ into $\mathscr{K}(U)$ is continuous.

Let $f\in \mathscr{K}(U)$. There exists a sequence $(f_m)_{m\in\mathbf{N}}$ in $\mathscr{D}(U)$ which converges uniformly to $f$ on U and such that the supports of the $f_m$ are contained in a fixed relatively compact neighbourhood of the support of $f$ (Lemma 3). The sequence $(f_m)$ therefore converges to $f$ in $\mathscr{K}(U)$. This concludes the proof of a).

Let $p\in [1,+\infty [$. Let $f\in \mathscr{D}(U)$ and let K be the support of $f$. One then has the inequality $N_p(f)\leqslant \mu(K)^{1/p}$ sup$_{x\in K}|f(x)|$, hence the canonical injection of $\mathscr{D}(U)$ into $\mathscr{L}^p(U)$ is continuous. The last part of assertion b) then results from a).

Since the product of two test functions is again a test function, Leibniz's formula (FVR, I, p. 28, prop. 2) shows that the space $\mathscr{D}(U)$ is a topological algebra.

More generally, let $f\in \mathscr{C}^{\infty}(U)$. The linear mapping $\varphi \mapsto f \varphi$ of $\mathscr{D}(U)$ into $\mathscr{D}(U)$ is then continuous. In fact, for every compact subset K of U and every $\alpha \in \mathbf{N}^n$, one has

$0\leqslant \beta \leqslant \alpha$

(cf. loc. cit.).