Suppose I have some linear operators $X_1, \dots, X_n$ on $\mathbb{C}^r$ (i.e. $r \times r$ matrices) and some other operators $Y_1^\epsilon, \dots, Y_n^\epsilon$ which are deformations of the $X_i$, by which I mean I can bound the differences $$ \| X_i - Y_i^\epsilon \| $$ by a function of $\epsilon$ which goes to $0$ as $\epsilon \to 0$, where $\| \cdot \|$ is some norm.
I want to show that $$ \operatorname{tr} (Y_1^\epsilon \cdots Y_n^\epsilon) \to \operatorname{tr}(X_1 \cdots X_n) $$ as $\epsilon \to 0$. What type of norm $\| \cdot \|$ should I consider to show this limiting relationship on the traces?
I might be able to assume that the $Y_i^\epsilon$ and $X_i$ have the same eigenvalues for each $i$.
So since you are in a finite dimensional space, all the norms are equivalent. So yes, $\|X\|_{2} := (\mathrm{tr}(X^*X))^{1/2}$, proposed by Borchers, is a good norm. Other good norms are the trace norm $\|X\|_{1} := \mathrm{tr}(|X|)$ where $|X| = (X^*X)^{1/2}$, the operator norm $\|X\| = \|X\|_{\infty} = \sup{\{\text{eigenvalues}\}}$. In all these norms, you have Hölder's inequality $$ \mathrm{tr}(XY) ≤ \|X\|_2\|Y\|_2 \\ \mathrm{tr}(XY) ≤ \|X\|_1\|Y\|. $$ Then you can write $\mathrm{tr}(Y_1Y_2Y_3) = \mathrm{tr}((Y_1-X_1)Y_2Y_3)+\mathrm{tr}(X_1Y_2Y_3) = \dots = \mathrm{tr}((Y_1-X_1)Y_2Y_3)+\mathrm{tr}(X_1(Y_2-X_2)X_3)+\mathrm{tr}(X_1X_2(Y_3-X_3))+\mathrm{tr}(X_1X_2X_3)$ and you can use Hölder's inequality to get the convergence you need.