Every unitarily invariant matrix norm is sub-multiplicative?
In R. Bhatia, Matrix Analysis, after Proposition IV.2.4, it says that "Every unitarily invariant matrix norm is sub-multiplicative". But I could not verify...
Here, a norm $||\cdot||$ is called unitarily invariant if $$||UAV||=||A||$$ for all matrix $A$, and unitary matrix $U,V$. And a norm is called sub-multiplicative if $$||AB||\leq ||A|| \cdot ||B||.$$
You have to combine IV.38 and IV.40. The first inequality says that $\|\cdot\|_\infty\leq\|\cdot\|$, where $\|\cdot\|_\infty$ is the operator norm and $\|\cdot\|$ is any unitarily invariant norm. Then, by IV.40, $$ \|AB\|\leq\|A\|_\infty\,\|B\|\leq\|A\|\,\|B\|. $$