Chapter 3.10, Problem 6P

Write out the proof of the Schwarz inequality $\left(10.9\right)$ for a complex Euclidean space. Hint: Follow the proof of $\left(10.4\right)$ in $\left(10.5\right),$ replacing the definitions of norm and inner product in $\left(10.1\right)$ and $\left(10.2\right)$ by the definitions in $\left(10.6\right)$ and $\left(10.7\right).$ Remember that norms are real and $\ge 0.$