Cauchy Bunyakovsky Schwarz inequality

We shall prove that for real $ x_i,y_i $ we have

$$ \left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right). $$

Function f below is always positive and has complex or double real roots.

$$f(t)=    (x_1 t + y_1)^2 + \cdots + (x_n t + y_n)^2. =\left (\sum_{i=1}^n x_i^2 \right )t^2+2\left (\sum_{i=1}^n x_i y_i \right)t+\left( \sum_{i=1}^n y_i^2\right)\ge 0$$

Then

$$\Delta=4\left(\sum_{i=1}^n x_i y_i\right)^2- 4\left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).\leq 0 $$

Back to top