Smith's lemma

For a matricially normed space $X$ and a linear operator $\Phi:X\to M_n$, we have $\Vert\Phi\Vert _{\mathrm{cb}} = \Vert\Phi^{(n)}\Vert$. In particular, $\Phi$ is completely bounded if and only if $\Phi^{(n)}$ is bounded [Smi83, Thm. 2.10].