Induced operator norms make M_n(R) a Banach algebra

Idea. The proofs for the Euclidean operator norm used only the definition as a supremum; they carry over verbatim. Equivalence of all norms in finite dimension gives finiteness and the comparison.

Step 0: finiteness. By equivalence of norms on $\mathbb{R}^n$, there are $0<c\le C$ with $c|x|\le|x|'\le C|x|$ for all $x$. Then $|Ax|'\le C|Ax|\le C\|A\||x|\le\frac Cc\|A\||x|'$, so $\|A\|'\le\frac Cc\|A\|<\infty$.

Step 1: norm axioms.
- Homogeneity and triangle inequality: the supremum of the seminorms $A\mapsto|Ax|'$ is a seminorm.
- Definiteness: if $\|A\|'=0$ then $Ax=0$ for all $x$, so $A=0$.
- $\|I\|'=\sup|x|'=1$.
- Submultiplicativity: as for the Euclidean case, $|ABx|'\le\|A\|'|Bx|'\le\|A\|'\|B\|'|x|'$.

Step 2: comparison and completeness. By Step 0, $\|A\|'\le\frac Cc\|A\|$. Conversely, $|Ax|\le\frac1c|Ax|'\le\frac1c\|A\|'|x|'\le\frac Cc\|A\|'|x|$, so $\|A\|\le\frac Cc\|A\|'$. This is part 3. Since $\|\cdot\|'$ is equivalent to $\|\cdot\|$ and $M_n$ is complete for $\|\cdot\|$, it is complete for $\|\cdot\|'$ (part 2).

Examples. $\|\cdot\|_1$ (max column sum) and $\|\cdot\|_\infty$ (max row sum) are induced by the $\ell^1$ and $\ell^\infty$ norms. The Frobenius norm is *not* an induced norm for $n\ge2$ since $\|I\|_F=\sqrt n\ne1$, though it is submultiplicative.

Remark. Part 3 says that every norm on $\mathbb{R}^n$ gives an operator norm equivalent to the Euclidean one. The same proof shows that in a Banach algebra the Neumann series and the openness of the group of invertible elements hold for any induced norm.