Let $A$ be finite-dimensional associative algebra over $\mathbb{R}$. What are the properties that must hold for $\|\cdot\|$ to be a norm on $A$? I assume we need:
- For all $x,y \in A$, $\|x+y\| \leq \|x\|+\|y\|$
- For all $a \in \mathbb{R}$, $x \in A$, $\|ax\| = |a|\|x\|$
- For all $x \in A$, $x=0$ if and only if $\|x\|=0$
I can't find the definition of a norm on an algebra anywhere, so I would appreciate a reference if possible.