The [[Operator Norm]] is indeed a [[Norm]]. ###### Proof Proof. The first two items are elementary and the proofs are left to the reader. For the third item, $\|(A+B) x\|=\|A x+B x\| \leqslant\|A x\|+\|B x\| \leqslant\left(\|A\|_{o p}+\|B\|_{o p}\right)\|x\|$and therefore $\|A+B\|_{o p}=\sup _{x \in \mathbb{R}^n \backslash\{0\}} \frac{\|(A+B) x\|}{\|x\|} \leqslant\|A\|_{o p}+\|B\|_{o p} .$