I have a question about Hopf duals. To begin we can remind the definition: For an Hopf algebra $A$ over a field $k$, the Hopf dual $A^{\circ}$ is the subspace of the linear dual $\mathrm{Lin}_k(A,k)$ consisting of elements that vanish on some two-sided ideal $I \subseteq A$ with finite codimension (i.e. $\mathrm{dim}(A/I) < \infty$). 

Let $B$ be a Hopf subalgebra of $A$. We can certainly consider the Hopf dual $B^{\circ}$. Is it true that any element of $B^{\circ}$ can be extended to an element of $A^{\circ}$?