Additivity of matrix norms?

Question

Let $A$, $B$, and $C$ be real-valued square matrices, and let $A = B + C$. What matrix norm, if any, satisfies $||A|| = ||B|| + ||C||$?

The Frobenius norm clearly does not satisfy this additivity requirement. Are there alternatives that do? If not, is there a proof that such additive matrix norms cannot exist?

Answer 1

No norm can satisfy this. If there is such a norm we get $0=\|0\|=\|A+(-A)\|=\|A\|+\|-A||=2\|A\|$ so $\|A\|=0$ for all $A$.