We can identify spaces $V$ and $V^{**}$ by canonical isomorphism: $$A:V\to V^{**},$$ $$Av(f)=f(v),$$ for any $f\in V^*$.
But why we cannot identify $V$ and $V^*$ by $e^{*}_{i}(e_j)=\delta_{ij}$ (I understand that after change the basis of $V$ operator $B: V\to V^*$ will be changed)? What means that spaces $V$ and $V^{**}$ are identical? How we can use it?