I need a proof for the Cauchy-Schwarz inequality on complex numbers, i.e.
$$|{(a_1b_1 + a_2b_2+…+a_nb_n)}|^2\leq(|a^2_1|+|a^2_2|+…+|a^2_n|) (|b^2_1|+|b^2_2|+…+|b^2_n|),$$ where each $a_i,b_i\in\mathbb{C}$. I thought of proving the $real$ and the $imaginary$ parts separately and then summing them up, but that will not work because on doing so the $real$ nos. formed by multiplying the $imaginary$ part of both numbers will not be considered. Please help me prove it.
Best Answer
There is no accepted definition of inequalities for complex numbers. The correct statement of Cauchy - Schwartz inequality is the the following: $\left\vert\ a_1\bar{b_1}+…+a_n\bar{b_n}\right\vert\leq \left(|a_1|^2+...+|a_1|^2\right) \left(|b_1|^2+...+ |b_1|^2 \right)$ and this follows immediately from the real case.