Here's Prob. 15, Chap. 1 in the book Principles of Mathematical Analysis by Wlater Rudin, 3rd edition:
Under what conditions does equality hold in the Schwarz inequality?
Now the Schwarz inequality, which is Theorem 1.35 in Rudin, is as follows:
If $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ are complex numbers, then $$ \left\vert \sum_{j=1}^n a_j \overline{b_j} \right\vert^2 \leq \sum_{j=1}^n \left\vert a_j \right\vert^2 \sum_{j=1}^n \left\vert b_j \right\vert^2.$$
If there is a complex number $z$ such that $a_j = z b_j$ for each $j=1, \ldots, n$, then we have $$ \begin{align} \left\vert \sum_{j=1}^n a_j \overline{b_j} \right\vert^2 &= \left\vert \sum_{j=1}^n z b_j \overline{b_j} \right\vert^2 \\ &= \left\vert z \ \sum_{j=1}^n \left\vert b_j \right\vert^2 \right\vert^2 \\ &= \vert z \vert^2 \cdot \left\vert \sum_{j=1}^n \left\vert b_j \right\vert^2 \right\vert^2 \\ &= \vert z \vert^2 \cdot \left( \sum_{j=1}^n \left\vert b_j \right\vert^2 \right)^2, \end{align} $$ whereas $$ \begin{align} \sum_{j=1}^n \left\vert a_j \right\vert^2 \sum_{j=1}^n \left\vert b_j \right\vert^2 &= \sum_{j=1}^n \left\vert z b_j \right\vert^2 \sum_{j=1}^n \left\vert b_j \right\vert^2 \\ &= \sum_{j=1}^n \vert z \vert^2 \left\vert b_j \right\vert^2 \sum_{j=1}^n \left\vert b_j \right\vert^2 \\ &= \vert z \vert^2 \cdot \left( \sum_{j=1}^n \left\vert b_j \right\vert^2 \right)^2 \\ &= \left\vert \sum_{j=1}^n a_j \overline{b_j} \right\vert^2. \end{align} $$
Now is this condition also a necessary condition for the equality to hold in the Schwarz "inequality"?