One defines the **finite dual** of a Hopf algebra $A$ as
$$
H^o := \{f \in H^* ~|~ f(I) = 0, \text{ for some ideal $I$ of $H$ with } \dim_C(H/I) < \infty \}.
$$
As is well-known, $H^o$ has a well-defined Hopf algebra structure obtained by dualizing the Hopf structure of $H$.

On the other hand, for any finite-dimensional $H$-module $V$, and element $v \in V$, and a functional $f \in V^*$,
we can define a functional $c_{f,v} \in H^*$ according to
$$
c_{f,v}(h) := f(hv).
$$
One usually calls any such functional a **matrix coefficient** of $H$. It is not difficult to see that the set of matrix coefficients
of $H$ forms a Hopf subalgebra of $H^o$, which we will denote by Mat$(H)$.

What I would like to know is when do we have the equality $$ H^o = \text{Mat}(H)? $$

5)in sites.google.com/site/darijgrinberg/hopfalgebren/… if you can read German (which I assume you can, given your location); it is currently on page 480. Sorry for the bad formatting... – darij grinberg Apr 6 '13 at 16:13