TITLE: Hopf dual of the Hopf dual
QUESTION [16 upvotes]: Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^*$ of as follows: Let $A^∗$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some two-sided ideal of $A$ of finite codimension. Then $A^∗$ has a natural Hopf algebra structure.
Question: Is the Hopf dual of the Hopf dual of $A$ isomorphic to $A$. It is not obvious for me that it is. If not, then do we know in which cases it is true?

REPLY [2 votes]: Regarding your first question: the answer is generally no, the restricted dual of the restricted dual of $A$ is generally not isomorphic to $A$:
$$
(A^{\circ})^\circ\ncong A
$$
as has already been indicated by the counterexamples of darij grinberg's answer. 
More counterexamples can be constructed when the restricted dual (or: Sweedler's dual) $A^\circ$ is trivial while $A$ is not:
Consider the case when the algebra $A$ has no finite dimensional representations, except the zero vector space $V=\{0\}$. Now, recall that 

$f\in A^\circ$ $\Longleftrightarrow$ $\dim(A\rightharpoonup f)<\infty$

where the action of the $A\rightharpoonup f$ module is defined by $(a\cdot f)(b)=f(ba)$ for all $a,b\in A$ and $f\in A^\circ$ (this is a standard result, see for example Montgomery's book, Lemma 9.1.1, p. 149). Thus, 

If $A$ has no finite dimensional representations apart from the zero vector space $V=\{0\}$, then $A^\circ=\{0\}$. 

Regarding your second question, unfortunately i have no examples or conditions available (and i think it is a difficult task to find something similar in the literature).