Suppose $u_1$ and $u_2$ are elements of $\mathbb{C}^N$ of norm $1$, and that $\langle u_1,u_2\rangle\not =0$. If $z$ is in $\mathbb{B}_N$ (the unit ball of $\mathbb{C}^N$), how many solutions does the equation $\langle z,u_1\rangle \langle z,u_2\rangle =\langle u_1,u_2\rangle $ have in $\mathbb{B}_N$?
Using the Cauchy-Schwarz inequality, the answer when $\big|\langle u_1,u_2\rangle\big|=1$ is that there are no solutions in $\mathbb{B}_N$. I am not sure how to proceed when $0<\big|\langle u_1,u_2\rangle\big|<1$.
Thank you.