Suppose $V$ is a finite dimensional vector space, $V'$ is its dual space, and $\{\phi_1, \dots, \phi_n\}$ is a basis for $V'$. Suppose I have found $\{v_1, \dots, v_n\}$ with $v_i \in V$ and that $\phi_i(v_j) = \delta_{ij}$ for $1 \leq i, j \leq n$. Is $\{v_1, \dots, v_n\}$ necessarily a basis for $V$?
(Note: I think YES because the double dual basis exists and is unique, and then $V''$ is canonically isomorphic to $V$. But, I want to be sure. Thanks in advance!)