Tightness of inequalities for various matrix norms For a general inequality involving matrix norms, does the choice of the norm influence the tightness of the inequality?
Eg. In $\|AB\| \leq \|A\| \|B\|$,
Does the choice of the norm affect the tightness of the inequality? Is there any particular norm in which inequalities are always tighter than other norms?
 A: An example of a matrix norm where the submultiplicative inequality is not tight (except when $A$ or $B$ is zero):
Taking $|\cdot|$ to be any norm (for example, take the spectral norm), we note that the norm $\|\cdot\|$ defined by $\|A\| = c|A|$ is another matrix norm for any $c > 1$.
We note that if $|AB| = |A| \cdot |B|$, then 
$$
\|AB\| = c\cdot|A| \cdot |B| = \frac 1c \cdot \|A\| \cdot \|B\| < \|A\| \cdot \|B\|
$$
So, we can create a matrix norm that is only tight when $A = 0$ or $B = 0$.

For most norms (i.e. for any I can think of, though I'm not sure that this is guaranteed), this inequality is tight in the sense that we can choose some non-zero matrices $A,B$ such that
$$
\|AB\| = \|A\|\cdot \|B\|
$$
However, here's a result that seems to be at least in the spirit of your question: let $\|\cdot\|$ denote any unitarily invariant norm, and let $|\cdot|$ denote the spectral norm.  We always have $|A| \leq \|A\|$, and moreover,
$$
\|ABC\| \leq |A|\cdot\|B\|\cdot |C|
$$
Among the Schatten $p$-norms, we also have Hölder's inequality, which is to say
$$
\|AB\|_1 \leq \|A\|_p \|A\|_q
$$
