In Linear Algebra Done Right (2nd edition), in the proof of Theorem 6.45, it is written:
Let $(e_1,...,e_n)$ be an orthonormal basis of $V$. Then $\psi(u)=\psi(\left<u,e_1\right>e_1+\dots+\left<u,e_n\right>e_n)=\left<u,e_1\right>\psi(e_1)+\dots+\left<u,e_n\right>\psi(e_n)=\left<u,\overline{\psi(e_1)}e_1+\dots+\overline{\psi(e_1)}e_1\right>$.
I understand that the first equality comes from 6.17, and the second equality comes from the fact that $\psi$ is a linear functional, and thus is a linear map. How do we get the third equality?