= Solution
Use the convention that the <Hermitian inner product> is conjugate-linear in its first argument and linear in its second. In <orthonormal coordinates>, $\langle v,w\rangle=\sum_i\overline{v_i}w_i$. It satisfies <conjugate symmetry>, sesquilinearity and positive definiteness. Expanding in the <orthonormal basis> gives
$$
v=\sum_i e_i\langle e_i,v\rangle,\qquad
w=\sum_j e_j\langle e_j,w\rangle.
$$
Applying the <Hermitian inner product> and $\langle e_i,e_j\rangle=\delta_{ij}$ yields
$$
\boxed{\langle v,w\rangle=\sum_i\overline{\langle e_i,v\rangle}\langle e_i,w\rangle
=\sum_i\langle v,e_i\rangle\langle e_i,w\rangle.}
$$
This identifies exactly where <complex conjugation> enters; replacing the first factor by $\langle e_i,v\rangle$ would generally be wrong under this convention.
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