One defines the finite dual of a Hopf algebra $H$ 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 $\operatorname{Mat}(H)$.
What I would like to know is when do we have the equality $$ H^o = \operatorname{Mat}(H)? $$
