Q:

Definition of annihilator is not clear

I have a question regarding the following definition of an annihilator of a finite dimensional vector space. I think I understand the two definitions but I don't really get the link implied by the last sentence. 
Definition:
If $A \subset V$, the annihilator of $A, A^{°}$, is the set of all $f$ in $V^*$ such that $f(a) = 0$ for all $a$ in $A$. Similarly, if $A \subset V^*$, then
$$
A^{°} = \{a\in V\colon\ f(a) = 0 \text{ for all } f\in A \}.
$$
If we view $V$ as $(V^{*})^{*}$, the second definition is included in the first.
So now my questions are why may we view $V$ as $(V^{*})^{*}$, I know they're equivalent by the isomorphism but this doesn't mean that they're equal, does it? And how does the first definition include the second one?

A:

They are not equal, but algebraically isomorphic, because a finite dimensional vector space is reflexive and the second definition is included in the first if You identify every $a\in V$ with it´s image $Ja$ under the canonical injection (in this case isomorphism) $J:V\rightarrow (V^*)^*,a\mapsto(Ja:V^*\rightarrow\mathbb{C},f\mapsto f(a))$,because the first definition then reads for $V^*$:
$ A^0=\{Ja\in V^{**}:Ja(f)=f(a)=0\}$

