I've started reading about distributions in Folland's Real Analysis, and on page 282 the book says the following:
Recall that for $E \subset \mathbb{R}^n$ we have defined $C_c^{\infty}(E)$ to be the set of all $C^{\infty}$ functions whose support is compact and contained in $E$. If $U \subset \mathbb{R}^n$ is open, $C_c^{\infty}(U)$ is the union of the spaces $C_c^{\infty}(K)$ as $K$ ranges over all compact subsets of $U$. Each of the latter is a Frechet space with the topology defined by the norms $$ \phi \mapsto \|\partial^{\alpha} \phi \|_u \qquad (\alpha \in \{0,1,2,\ldots,\}^n), $$
in which a sequence $\{\phi_j \}$ converges iff $\partial^{\alpha} \phi_j \to \partial^{\alpha} \phi$ uniformly for all $\alpha$. (The completeness of $C_c^{\infty}(K)$ is easily proved by the argument in Exercise 9 in $\S 5.1$.)
From what I understand, the notion of completeness requires a norm. If that's the case, then what exactly is the norm with which we are equipping $C_c^{\infty}(K)$? Would it be
$$ \|\phi\| := \sum_{\alpha} \| \partial^{\alpha} \phi \|_u = \sum_{k=0}^{\infty} \sum_{|\alpha| = k} \| \partial^{\alpha} \phi \|_u \quad ? $$