$\mathcal S(\mathbb R^n) := \{f \in C^{\infty}(\mathbb R^n): \|\varphi\|_{k,l} < \infty \}$ where $\|\varphi\|_{k,l} := \operatorname{sup}_{x \in \mathbb R^n} (1 + |x|^2)^{k \over 2} \sum_{|\alpha| \le l} |D^{\alpha}\varphi(x)|, \alpha \in \mathbb N^n_0$. We write $\varphi_j \overset{\mathcal S}{\to} \varphi$, if $\lim_{j \to \infty}|\varphi_j - \varphi\|_{k,l} \to 0 $ $\forall k,l \in \mathbb N$.
Then, $T: \mathcal S(\mathbb R^n) \to \mathbb C \in \mathcal S'(\mathbb R^n)$, if $T$ is linear and $T\varphi_j \to T\varphi$ for all $\varphi_j \overset{\mathcal S}{\to} \varphi$.
Q: What kind of convergence is $T\varphi_j \to T\varphi$ in $\mathbb C$? Is it with respect to the usual (euclidean) norm?
Yes, it is of course with respect to the usual norm in ${\bf{C}}$ that $|x+iy|=\sqrt{x^{2}+y^{2}}$. But you know, in Euclidean space, all norms are equivalent.
And the continuity of $T$ is with respect to the topological vector space $S({\bf{R}}^{n})$ and the usual topology of ${\bf{C}}$.
Note that we have some sort of boundedness characterization of the continuity of $T$: $|T(\phi)|\leq C_{\alpha,\beta,n}\displaystyle\sum_{|k|\leq\alpha,|l|\leq\beta}\rho_{k,l}(f)$, where $\rho_{k,l}$ are those seminorms.