All norms on M_n(R) are equivalent

Idea. The Frobenius unit sphere is compact (closed and bounded in $\mathbb{R}^{n^2}$) and any norm is continuous and positive on it, so it has a positive minimum and a finite maximum. Homogeneity extends the bounds to all matrices.

Step 1: upper bound. Write $A=\sum a_{ij}E_{ij}$. By the triangle inequality and Cauchy-Schwarz,
$$\|A\|'\le\sum_{i,j}|a_{ij}|\|E_{ij}\|'\le\Big(\sum_{i,j}\|E_{ij}\|'^2\Big)^{1/2}\|A\|_F=:C\|A\|_F.$$

Step 2: continuity. By the reverse triangle inequality, $\big|\|A\|'-\|B\|'\big|\le\|A-B\|'\le C\|A-B\|_F$: the function $\|\cdot\|'$ is continuous for the Frobenius topology.

Step 3: lower bound. The Frobenius unit sphere $S=\{\|A\|_F=1\}$ is compact. The continuous function $\|\cdot\|'$ attains its minimum $c$ on $S$ at some $A_0\in S$; and $c=\|A_0\|'>0$ since $A_0\ne0$.

Step 4: homogeneity. For $A\ne0$, $A/\|A\|_F\in S$, hence $c\le\|A/\|A\|_F\|'=\|A\|'/\|A\|_F$, i.e. $c\|A\|_F\le\|A\|'$. Together with Step 1 we get $c\|A\|_F\le\|A\|'\le C\|A\|_F$, and $c\le C$ is automatic.

Remark. The compactness of the unit sphere is where finite dimension enters; in infinite dimension the unit sphere is not compact and the statement fails. The result implies that all the topological notions (open, closed, compact, convergence, continuity of maps between $M_n$'s) do not depend on the chosen norm, which is why we freely use whichever norm is most convenient.