If we have a basis $\{f_1,...,f_n\}$ of $V^*$, can we give a basis $\{x_1,...,x_n\}$ of $V$? Let $V$ be a vector space of dimension $n$. I know that if $\{x_1,...,x_n\}$ is a basis of $V$ then we can give easily a basis $\{f_1,...,f_n\}$ of the dual space $V^*$ by setting $f_i(x_j)=\delta_{ij}$.
My question now is:

If we have a basis $\{f_1,...,f_n\}$ of $V^*$, is it possible to give a basis $\{x_1,...,x_n\}$ of $V$ such that $f_i(x_j)=\delta_{ij}$?

What do you think?
Thank you.

Yes, in finite dimension, $V$ is $(V^*)^*$ that this the bidual of $V$ is $V$.
