In Laurent Schwartz book on distribution theory he defines a family of semi-norms that generates the topology on $C_c^{\infty}(\Omega), \ \Omega \subset \mathbb{R}^n$ : $N(\{m\}, \{ \epsilon \} )(\phi) = \sup_{\nu} \sup_{|x|\geq \nu, \ |p|\leq m_\nu} |D^p \phi(x) | \frac{1}{\epsilon_\nu}$ where $\{ m\} =\{m_1, \dots m_\nu \dots\}$ is increasing to $+\infty$ and $\{\epsilon\} =\{ \epsilon_1, \dots, \epsilon_\nu , \dots\}$ is decreasing to 0.
If I understand it right what one wants to achieve with these semi norms is to squeeze Cauchy sequences to the same compact subset $K \subset \Omega$ but I don't really see how they do this? Why do you look at $|x|\geq \nu$? Then they wont necessarily converge once they're in the same set $K$, or what am I missing?
Also I think they're restriction to $K$ should coincide with the semi-norms that generate $\mathcal{D}_K$; $\sup_{k\leq n, x\in K} |\phi(x)|$, but I don't really see how this is the case? Any help is much appreciated!
Maybe that's what he meant :
Let $a_n \to \infty$. If $\sup_n \sup_{|x| \ge n} a_n |f(x)|$ is bounded then $\sup_{|x| \ge n} |f(x)| = \mathcal{O}(1/a_n)$.
If for every sequence $(a_n)$ : $\quad\sup_n \sup_{|x| \ge n} a_n |f(x)|$ is bounded then $f$ has compact support.
Doing the same with $\sup_{p \le k} |f^{(p)}(x)|$ means $f \in C^\infty_c$.
It is not so useful, except that $\|f\|_{a,k} = \sup_n \sup_{|x| \ge n} a_n \sum_{p \le k}|f^{(p)}(x)|$ is a norm on some Banach space $B_{p,a}$, thus $C^\infty_c = \bigcap B_{p,a}$ is complete for the topology induced by those norms.