Goal: Let $u_i$ and $u_j$ be the $i$th and $j$th columns of unitary matrix $U$, respectively. We wish to show that
$$ \langle u_i, u_j \rangle = 0, i \ne j \\ \langle u_i, u_j \rangle = 1, i = j \\ $$
From pg. 28 of Principles of Quantum Mechanics, the author presents two proofs of this, the second of which confuses me:
Question: Why does showing that
$$ \sum_k U_{ki}^\star U_{kj} = \delta_{ij} $$
demonstrate the goal? Here it seems that the author is equating
$$ \langle u_i, u_j \rangle $$
with the complex dot product
$$ u_i \cdot u_k = \sum_k \overline{{u_i}_k} {u_j}_k = \sum_k U_{ki}^* U_{kj} $$
but aren't those not necessarily always the same thing?